The document discusses properties of derivatives and how they relate to limits. It states that the sum, difference, and constant multiple rules for limits directly apply to differentiation. However, the product and quotient rules for limits do not directly apply to differentiation, which has more complicated product and quotient rules. Elementary functions are defined in terms of a few basic formulas and operations. The document then examines the sum and constant multiple rules for derivatives in more detail, proving them using limits. It also provides a geometric illustration of how the derivative of a sum is equal to the sum of the derivatives.
The document discusses limits of fractional expressions as the variable approaches certain values. It provides four basic facts about the limits of fractions of elementary functions: (1) if the numerator and denominator have defined limits, the fractional limit is the fraction of the limits; (2) if the numerator is bounded and the denominator diverges, the fractional limit is 0; (3) if the numerator diverges and the denominator is bounded, the fractional limit is infinity; (4) if both the numerator and denominator have limits of 0 or infinity, the fractional limit is inconclusive. It emphasizes that an undefined fractional limit does not necessarily mean the limit is inconclusive - it may simply not exist. Rationalizing expressions can sometimes resolve inconclusive fractional limits
1.2 review on algebra 2-sign charts and inequalitiesmath265
The document discusses sign charts and inequalities. It explains that sign charts can be used to determine if expressions are positive or negative by factoring them and evaluating at given values of x. Examples are provided to demonstrate how to construct a sign chart by: 1) solving for where the expression equals 0, 2) marking these values on a number line, and 3) evaluating the expression at sample points in each segment to determine the signs in between values where the expression equals 0. The sign chart then indicates the ranges where the expression is positive, negative or zero.
The document discusses exponential and logarithmic functions. Exponential functions of the form f(x) = b^x are called exponential functions in base b. Logarithmic functions log_b(y) represent the exponent x needed to raise the base b to a power to get the output y. The exponential form b^x = y and logarithmic form x = log_b(y) describe the same relationship between the base b, exponent x, and output y. Questions can be translated between these forms by rewriting the exponential expression as a logarithm or vice versa. Examples demonstrate rewriting expressions and graphing logarithmic functions.
3.3 graphs of factorable polynomials and rational functionsmath265
The document discusses graphs of factorable polynomials. It begins by showing examples of graphs of even and odd degree polynomials like y=x2, y=x4, y=x3, and y=-x5. It then explains that the graphs of polynomials are smooth, unbroken curves. For large values of x, the leading term of a polynomial dominates and determines the graph's behavior. Based on the leading term and whether the degree is even or odd, the graph exhibits one of four behaviors as x approaches infinity. The document demonstrates how to construct the sign chart of a polynomial from its roots and use it to sketch the central portion of the graph. It provides an example of sketching the graph of y=x
This document discusses rules for computing derivatives of functions. It begins by listing existing derivative rules and defining notation. It then derives and presents rules for the derivatives of trigonometric functions like sine, cosine, tangent, cotangent, secant and cosecant. An example problem demonstrates finding the derivative of the tangent function using previous rules.
3.2 implicit equations and implicit differentiationmath265
The document discusses implicit equations and implicit differentiation. It begins by explaining the difference between explicit and implicit forms of equations, using the example of y=1/x which can be written explicitly as y=1/x or implicitly as xy=1. It then introduces the concept of implicit differentiation, which involves taking the derivative of an implicit equation with respect to x and solving for the derivative of y with respect to x (y’). This allows one to find the slope of the curve at a point, even if the explicit form of the relation between x and y is difficult to determine from the implicit equation.
This document discusses optimization problems in real-world applications and the role of derivatives. It provides examples of functions that may or may not have extrema over an interval. The extrema theorem for continuous functions states that a continuous function over a closed interval will have both an absolute maximum and minimum. Extrema can occur where the derivative is zero, where the derivative is undefined, or at the endpoints. Examples are provided to illustrate the different types of extrema.
There are two systems for measuring angles: the degree system and the radian system. The degree system divides a full circle into 360 equal angles of 1 degree each. The radian system defines an angle as the arc length cut out by the angle on a unit circle of radius 1, where a full circle corresponds to 2π radians. While the degree system is commonly used, the radian system is preferred in mathematics due to its relationship to circle geometry formulas involving arc lengths and wedge areas.
The document discusses limits of fractional expressions as the variable approaches certain values. It provides four basic facts about the limits of fractions of elementary functions: (1) if the numerator and denominator have defined limits, the fractional limit is the fraction of the limits; (2) if the numerator is bounded and the denominator diverges, the fractional limit is 0; (3) if the numerator diverges and the denominator is bounded, the fractional limit is infinity; (4) if both the numerator and denominator have limits of 0 or infinity, the fractional limit is inconclusive. It emphasizes that an undefined fractional limit does not necessarily mean the limit is inconclusive - it may simply not exist. Rationalizing expressions can sometimes resolve inconclusive fractional limits
1.2 review on algebra 2-sign charts and inequalitiesmath265
The document discusses sign charts and inequalities. It explains that sign charts can be used to determine if expressions are positive or negative by factoring them and evaluating at given values of x. Examples are provided to demonstrate how to construct a sign chart by: 1) solving for where the expression equals 0, 2) marking these values on a number line, and 3) evaluating the expression at sample points in each segment to determine the signs in between values where the expression equals 0. The sign chart then indicates the ranges where the expression is positive, negative or zero.
The document discusses exponential and logarithmic functions. Exponential functions of the form f(x) = b^x are called exponential functions in base b. Logarithmic functions log_b(y) represent the exponent x needed to raise the base b to a power to get the output y. The exponential form b^x = y and logarithmic form x = log_b(y) describe the same relationship between the base b, exponent x, and output y. Questions can be translated between these forms by rewriting the exponential expression as a logarithm or vice versa. Examples demonstrate rewriting expressions and graphing logarithmic functions.
3.3 graphs of factorable polynomials and rational functionsmath265
The document discusses graphs of factorable polynomials. It begins by showing examples of graphs of even and odd degree polynomials like y=x2, y=x4, y=x3, and y=-x5. It then explains that the graphs of polynomials are smooth, unbroken curves. For large values of x, the leading term of a polynomial dominates and determines the graph's behavior. Based on the leading term and whether the degree is even or odd, the graph exhibits one of four behaviors as x approaches infinity. The document demonstrates how to construct the sign chart of a polynomial from its roots and use it to sketch the central portion of the graph. It provides an example of sketching the graph of y=x
This document discusses rules for computing derivatives of functions. It begins by listing existing derivative rules and defining notation. It then derives and presents rules for the derivatives of trigonometric functions like sine, cosine, tangent, cotangent, secant and cosecant. An example problem demonstrates finding the derivative of the tangent function using previous rules.
3.2 implicit equations and implicit differentiationmath265
The document discusses implicit equations and implicit differentiation. It begins by explaining the difference between explicit and implicit forms of equations, using the example of y=1/x which can be written explicitly as y=1/x or implicitly as xy=1. It then introduces the concept of implicit differentiation, which involves taking the derivative of an implicit equation with respect to x and solving for the derivative of y with respect to x (y’). This allows one to find the slope of the curve at a point, even if the explicit form of the relation between x and y is difficult to determine from the implicit equation.
This document discusses optimization problems in real-world applications and the role of derivatives. It provides examples of functions that may or may not have extrema over an interval. The extrema theorem for continuous functions states that a continuous function over a closed interval will have both an absolute maximum and minimum. Extrema can occur where the derivative is zero, where the derivative is undefined, or at the endpoints. Examples are provided to illustrate the different types of extrema.
There are two systems for measuring angles: the degree system and the radian system. The degree system divides a full circle into 360 equal angles of 1 degree each. The radian system defines an angle as the arc length cut out by the angle on a unit circle of radius 1, where a full circle corresponds to 2π radians. While the degree system is commonly used, the radian system is preferred in mathematics due to its relationship to circle geometry formulas involving arc lengths and wedge areas.
The document discusses derivatives and graphs. It defines interval notation used to indicate whether points are included or excluded from intervals. It then explains that the derivative of a function f(x) at a point x, f'(x), represents the slope of the tangent line to the graph of f(x) at (x, f(x)). Finally, it notes that points where the derivative is 0 are called critical points, as the tangent line is flat at these points.
The document defines the derivative of a function f(x) as the limit of the difference quotient (f(x+h) - f(x))/h as h approaches 0. This represents the slope of the tangent line to the function f(x) at the point x. An example is worked out where the derivative of the function f(x) = x^2 - 2x + 2 is calculated to be 2x - 2. The derivative is denoted by f'(x) and represents the instantaneous rate of change of the function at the point x.
1) The document discusses derivatives as rates of change, using the example of a stone thrown straight up.
2) It is found that the stone will stay in the air for 6 seconds, reaching its maximum height of 144 feet after 3 seconds.
3) The derivative of the height function D(t) represents the instantaneous rate of change of height, or speed, at each time t. This rate varies throughout the stone's trajectory.
- The derivative of a function f(x) represents the instantaneous rate of change of the output y with respect to the input x. It is equivalent to the slope of the tangent line and the amount of change in y for a 1 unit change in x.
- For a linear price-demand function of y = f(x) chickens sold given price x, the derivative of the revenue function R(x) = x*f(x) represents how revenue changes with a 1 unit change in price.
- The price that maximizes revenue occurs when the derivative of the revenue function R'(x) is 0, as this is where revenue is no longer increasing or decreasing with small changes in price.
This document discusses algebraic expressions and polynomial expressions. It provides examples of algebraic expressions and defines them as formulas constructed with variables and numbers using basic arithmetic operations. Polynomials are defined as expressions of the form anxn + an-1xn-1 + ... + a1x + a0, where the ai's are numbers. The document gives examples of factoring polynomials and evaluating polynomial expressions at given values. It also discusses using factoring to find the roots of polynomial equations.
The document discusses the concept of slope and the difference quotient formula for calculating slope. It defines a function f(x) and points P(x,f(x)) and Q(x+h, f(x+h)) on the graph of f(x). The slope of the cord connecting points P and Q is given by the difference quotient (f(x+h) - f(x))/h. An example problem calculates this slope for the specific points P(2,2) and Q(2.2,2.44) on the parabola y=x^2 - 2x + 2.
The document discusses limits and derivatives. It explains that in calculating the derivative of f(x)=x^2 - 2x + 2, the slope formula was simplified. As h approaches 0, the chords slide towards the tangent line, so the slope at (x,f(x)) is 2x-2. It then provides definitions and explanations for what it means for a variable to approach 0 from the right, left, or in general, to clarify the procedure of obtaining slopes using limits.
The document discusses continuity of functions and graphs. It defines a continuous function as one where the graph is unbroken within its domain. A function is discontinuous if its graph is broken. Continuity at a point x=a can be determined by comparing the left and right limits of the function at a to the actual value of the function at a. If the limits are equal to the function value, it is continuous from that side. The document provides examples of functions that are right continuous, left continuous, or discontinuous at various points to illustrate these concepts.
The document discusses evaluating the formula log[(2x+1)/(sin1/3(x)+1)] at x=0 and x=10 degrees using a scientific calculator. It explains that the answer is 0 at x=0 and approximately 1.13 at x=10 degrees. It then describes the keyboard of a typical scientific calculator, noting the number, operation, yx, sin, log and formula keys. The rest of the document provides examples and definitions of algebraic, trigonometric and exponential-log formulas.
5.3 areas, riemann sums, and the fundamental theorem of calaculusmath265
The document defines definite integrals and Riemann sums. It states that a definite integral calculates the area under a function between limits a and b by dividing the interval into subintervals and summing the areas of rectangles approximating the function over each subinterval. Riemann sums make this approximation explicit by taking the width of each subinterval times the value of the function at a sample point in the subinterval. In the limit as the subintervals approach zero width, the Riemann sum converges to the true integral value.
The document discusses slopes and derivatives. It defines slope as the ratio of the "rise" over the "run" between two points on a line. For a curve, the slope at a point is defined as the slope of the tangent line at that point. The derivative at a point is also called the slope of the tangent line and represents the instantaneous rate of change of the function at that point. The document provides an example of using slopes to calculate rates like velocity and fuel efficiency from distance and time measurements.
The document discusses related rates problems. It begins by using resizing a rectangle on a computer screen as an example to demonstrate how the rates of change of the length (L) and width (W) relate to the rate of change of the area (A). The key steps are: (1) the area A is given by A=LW, (2) take the derivative of both sides, (3) use the product rule and chain rule to obtain A'=L'W+LW', (4) plug in the given rates of L' and W' to solve for A'.
The document then provides examples to demonstrate how to set up and solve related rates problems by translating the given rates into derivatives, applying
4.5 continuous functions and differentiable functionsmath265
The document discusses continuous and differentiable functions. It defines elementary functions as those constructed using basic operations like addition and multiplication. Continuous functions over a closed interval are bounded and have absolute maximum and minimum values. The Intermediate Value Theorem states that a continuous function takes on all values between its minimum and maximum. Differentiable functions are continuous. Rolle's Theorem says that if a differentiable function is equal at the endpoints of an interval, its derivative is zero somewhere in between.
The document summarizes different types of derivatives. It discusses simple derivatives where there is one input and output, and defines them. It then discusses implicit derivatives where a relationship between two variables is given and the derivative of one with respect to the other is sought using implicit differentiation. An example finds the derivative of u with respect to v and v with respect to u for the equation 2u^2 - v^3 = 2 - uv. Reciprocal relationships between the derivatives are noted.
The document provides an overview of the concept of derivatives. It states that a function is differentiable at a point if the slope of its tangent line at that point is well-defined. It also notes that a function is differentiable over an interval if it is differentiable at every point in the interval. The document then discusses how derivatives can be systematically calculated by taking the derivatives of basic functions like power, trigonometric, logarithmic and exponential functions, and understanding how derivatives behave under operations like addition, subtraction, multiplication, division and function composition.
The document discusses using the second derivative to identify extrema and classify flat points on a graph of y=f(x). It defines terms for the second derivative, explaining that if f''(x)>0, the slope f'(x) is increasing, meaning a downhill point is getting less steep and an uphill point is getting more steep. For a maximum point M, the curve must flatten out with f'(x) approaching 0+ and f'(x) becoming increasingly negative after M, resulting in f''(M)<0.
- The order of a root of a polynomial is the number of times the root repeats.
- The polynomial x5 + 2x4 + x3 has two roots, x = 0 with order 3 and x = -1 with order 2.
- In general, polynomials of the form k(x - c1)m(x - c2)m...(x - cn)m have roots x = c1 with order m1, x = c2 with order m2, and so on.
The document discusses calculating the slope of a curve between two points (x, f(x)) and (x+h, f(x+h)) using the difference quotient formula. It defines the difference quotient as (f(x+h) - f(x))/h, where h is the difference between x and x+h. An example calculates the slope between the points (2, f(2)) and (2.2, f(2.2)) for the function f(x) = x^2 - 2x + 2, finding the slope to be 0.44.
The document discusses notation and algebra of functions. It defines a function as a procedure that assigns a unique output to each valid input. Most mathematical functions are represented by formulas like f(x) = x^2 - 2x + 3, where f(x) is the name of the function, x is the input variable, and the formula defines the relationship between input and output. New functions can be formed using basic operations like addition, subtraction, multiplication, and division of existing functions. Examples are provided to demonstrate evaluating functions at given inputs and combining functions algebraically.
The document discusses various transformations that can be performed on graphs of functions, including vertical translations, stretches, and compressions. Vertical translations move the entire graph up or down by adding or subtracting a constant to the function. Stretches elongate or compress the graph vertically by multiplying the function by a constant greater than or less than 1, respectively. These transformations can be represented by modifying the original function in a way that corresponds to the geometric transformation of its graph.
The document discusses various chain rules for derivatives, including:
- The power chain rule: [up]' = pup−1(u)'
- Trigonometric chain rules: [sin(u)]' = cos(u)(u)', [cos(u)]' = −sin(u)(u)'
- Examples are provided to demonstrate applying the chain rules to find derivatives of more complex functions like y = sin(x3) and y = sin3(x). Repeated application of the appropriate chain rule at each step is often required.
The document discusses derivatives and graphs. It defines interval notation used to indicate whether points are included or excluded from intervals. It then explains that the derivative of a function f(x) at a point x, f'(x), represents the slope of the tangent line to the graph of f(x) at (x, f(x)). Finally, it notes that points where the derivative is 0 are called critical points, as the tangent line is flat at these points.
The document defines the derivative of a function f(x) as the limit of the difference quotient (f(x+h) - f(x))/h as h approaches 0. This represents the slope of the tangent line to the function f(x) at the point x. An example is worked out where the derivative of the function f(x) = x^2 - 2x + 2 is calculated to be 2x - 2. The derivative is denoted by f'(x) and represents the instantaneous rate of change of the function at the point x.
1) The document discusses derivatives as rates of change, using the example of a stone thrown straight up.
2) It is found that the stone will stay in the air for 6 seconds, reaching its maximum height of 144 feet after 3 seconds.
3) The derivative of the height function D(t) represents the instantaneous rate of change of height, or speed, at each time t. This rate varies throughout the stone's trajectory.
- The derivative of a function f(x) represents the instantaneous rate of change of the output y with respect to the input x. It is equivalent to the slope of the tangent line and the amount of change in y for a 1 unit change in x.
- For a linear price-demand function of y = f(x) chickens sold given price x, the derivative of the revenue function R(x) = x*f(x) represents how revenue changes with a 1 unit change in price.
- The price that maximizes revenue occurs when the derivative of the revenue function R'(x) is 0, as this is where revenue is no longer increasing or decreasing with small changes in price.
This document discusses algebraic expressions and polynomial expressions. It provides examples of algebraic expressions and defines them as formulas constructed with variables and numbers using basic arithmetic operations. Polynomials are defined as expressions of the form anxn + an-1xn-1 + ... + a1x + a0, where the ai's are numbers. The document gives examples of factoring polynomials and evaluating polynomial expressions at given values. It also discusses using factoring to find the roots of polynomial equations.
The document discusses the concept of slope and the difference quotient formula for calculating slope. It defines a function f(x) and points P(x,f(x)) and Q(x+h, f(x+h)) on the graph of f(x). The slope of the cord connecting points P and Q is given by the difference quotient (f(x+h) - f(x))/h. An example problem calculates this slope for the specific points P(2,2) and Q(2.2,2.44) on the parabola y=x^2 - 2x + 2.
The document discusses limits and derivatives. It explains that in calculating the derivative of f(x)=x^2 - 2x + 2, the slope formula was simplified. As h approaches 0, the chords slide towards the tangent line, so the slope at (x,f(x)) is 2x-2. It then provides definitions and explanations for what it means for a variable to approach 0 from the right, left, or in general, to clarify the procedure of obtaining slopes using limits.
The document discusses continuity of functions and graphs. It defines a continuous function as one where the graph is unbroken within its domain. A function is discontinuous if its graph is broken. Continuity at a point x=a can be determined by comparing the left and right limits of the function at a to the actual value of the function at a. If the limits are equal to the function value, it is continuous from that side. The document provides examples of functions that are right continuous, left continuous, or discontinuous at various points to illustrate these concepts.
The document discusses evaluating the formula log[(2x+1)/(sin1/3(x)+1)] at x=0 and x=10 degrees using a scientific calculator. It explains that the answer is 0 at x=0 and approximately 1.13 at x=10 degrees. It then describes the keyboard of a typical scientific calculator, noting the number, operation, yx, sin, log and formula keys. The rest of the document provides examples and definitions of algebraic, trigonometric and exponential-log formulas.
5.3 areas, riemann sums, and the fundamental theorem of calaculusmath265
The document defines definite integrals and Riemann sums. It states that a definite integral calculates the area under a function between limits a and b by dividing the interval into subintervals and summing the areas of rectangles approximating the function over each subinterval. Riemann sums make this approximation explicit by taking the width of each subinterval times the value of the function at a sample point in the subinterval. In the limit as the subintervals approach zero width, the Riemann sum converges to the true integral value.
The document discusses slopes and derivatives. It defines slope as the ratio of the "rise" over the "run" between two points on a line. For a curve, the slope at a point is defined as the slope of the tangent line at that point. The derivative at a point is also called the slope of the tangent line and represents the instantaneous rate of change of the function at that point. The document provides an example of using slopes to calculate rates like velocity and fuel efficiency from distance and time measurements.
The document discusses related rates problems. It begins by using resizing a rectangle on a computer screen as an example to demonstrate how the rates of change of the length (L) and width (W) relate to the rate of change of the area (A). The key steps are: (1) the area A is given by A=LW, (2) take the derivative of both sides, (3) use the product rule and chain rule to obtain A'=L'W+LW', (4) plug in the given rates of L' and W' to solve for A'.
The document then provides examples to demonstrate how to set up and solve related rates problems by translating the given rates into derivatives, applying
4.5 continuous functions and differentiable functionsmath265
The document discusses continuous and differentiable functions. It defines elementary functions as those constructed using basic operations like addition and multiplication. Continuous functions over a closed interval are bounded and have absolute maximum and minimum values. The Intermediate Value Theorem states that a continuous function takes on all values between its minimum and maximum. Differentiable functions are continuous. Rolle's Theorem says that if a differentiable function is equal at the endpoints of an interval, its derivative is zero somewhere in between.
The document summarizes different types of derivatives. It discusses simple derivatives where there is one input and output, and defines them. It then discusses implicit derivatives where a relationship between two variables is given and the derivative of one with respect to the other is sought using implicit differentiation. An example finds the derivative of u with respect to v and v with respect to u for the equation 2u^2 - v^3 = 2 - uv. Reciprocal relationships between the derivatives are noted.
The document provides an overview of the concept of derivatives. It states that a function is differentiable at a point if the slope of its tangent line at that point is well-defined. It also notes that a function is differentiable over an interval if it is differentiable at every point in the interval. The document then discusses how derivatives can be systematically calculated by taking the derivatives of basic functions like power, trigonometric, logarithmic and exponential functions, and understanding how derivatives behave under operations like addition, subtraction, multiplication, division and function composition.
The document discusses using the second derivative to identify extrema and classify flat points on a graph of y=f(x). It defines terms for the second derivative, explaining that if f''(x)>0, the slope f'(x) is increasing, meaning a downhill point is getting less steep and an uphill point is getting more steep. For a maximum point M, the curve must flatten out with f'(x) approaching 0+ and f'(x) becoming increasingly negative after M, resulting in f''(M)<0.
- The order of a root of a polynomial is the number of times the root repeats.
- The polynomial x5 + 2x4 + x3 has two roots, x = 0 with order 3 and x = -1 with order 2.
- In general, polynomials of the form k(x - c1)m(x - c2)m...(x - cn)m have roots x = c1 with order m1, x = c2 with order m2, and so on.
The document discusses calculating the slope of a curve between two points (x, f(x)) and (x+h, f(x+h)) using the difference quotient formula. It defines the difference quotient as (f(x+h) - f(x))/h, where h is the difference between x and x+h. An example calculates the slope between the points (2, f(2)) and (2.2, f(2.2)) for the function f(x) = x^2 - 2x + 2, finding the slope to be 0.44.
The document discusses notation and algebra of functions. It defines a function as a procedure that assigns a unique output to each valid input. Most mathematical functions are represented by formulas like f(x) = x^2 - 2x + 3, where f(x) is the name of the function, x is the input variable, and the formula defines the relationship between input and output. New functions can be formed using basic operations like addition, subtraction, multiplication, and division of existing functions. Examples are provided to demonstrate evaluating functions at given inputs and combining functions algebraically.
The document discusses various transformations that can be performed on graphs of functions, including vertical translations, stretches, and compressions. Vertical translations move the entire graph up or down by adding or subtracting a constant to the function. Stretches elongate or compress the graph vertically by multiplying the function by a constant greater than or less than 1, respectively. These transformations can be represented by modifying the original function in a way that corresponds to the geometric transformation of its graph.
The document discusses various chain rules for derivatives, including:
- The power chain rule: [up]' = pup−1(u)'
- Trigonometric chain rules: [sin(u)]' = cos(u)(u)', [cos(u)]' = −sin(u)(u)'
- Examples are provided to demonstrate applying the chain rules to find derivatives of more complex functions like y = sin(x3) and y = sin3(x). Repeated application of the appropriate chain rule at each step is often required.
The document discusses higher order derivatives. It defines the nth derivative of a function f(x) as f(n)(x). The first example finds the first five derivatives of f(x)=2x^4 - x^3 - 2. The second example finds the first three derivatives of f(x)=-x^2/3. The third example finds the first four derivatives of f(x)=ln(x) and discusses how derivatives of rational functions become more complicated with higher orders. It also provides examples of finding derivatives of other functions like sin(x).
The document describes how to calculate the volume of a solid object using Cavalieri's principle. It involves partitioning the solid into thin cross-sectional slices and approximating the volume of each slice as a cylinder with the slice's cross-sectional area and thickness. The total volume is then approximated as the sum of the cylindrical slice volumes. As the number of slices approaches infinity, this sum approaches the actual volume calculated as the integral of the cross-sectional area function over the solid's distance range.
The document discusses antiderivatives and integration. It defines an antiderivative as a function whose derivative is the original function. The integral of a function is defined as the set of its antiderivatives. Basic integration rules are provided, such as integrating term-by-term and pulling out constants. Formulas for integrating common functions like exponentials, trigonometric functions, and logarithms are listed. An example problem demonstrates finding the antiderivative of a multi-term function by applying the basic integration rules.
The document discusses the substitution method of integration. It explains that while the derivative of an elementary function is another elementary function, the antiderivative may not be. There are two main integration methods: substitution and integration by parts. Substitution reverses the chain rule by letting u be a function of x with derivative u', then substituting u for x and replacing dx with du/u' in the integral.
This document discusses two applications of tangent lines: differentials and linear approximation, and finding the tangent line T(b) at a nearby point b. It explains that the tangent line T(x) at point (a, f(a)) is given by T(x) = f'(a)(x - a) + f(a). The slope f'(a) is identified with the derivative dy/dx. There are two ways to find T(b): directly using T(x), or by finding the differential ΔT = dy and using ΔT + f(a) = T(b).
2 integration and the substitution methods xmath266
The document discusses anti-derivatives and integration. It defines an anti-derivative F(x) of a function f(x) as any function whose derivative is f(x). It also calls F(x) the indefinite integral of f(x). The document outlines basic integration rules for adding, subtracting and multiplying terms by constants when taking integrals. It provides examples of integrals of common functions and explains that the integration constant k positions the graph of the anti-derivative F(x).
This document provides an overview of key calculus concepts including:
- Functions and function notation which are fundamental to calculus
- Limits which allow defining new points from sequences and are essential to calculus concepts like derivatives and integrals
- Derivatives which measure how one quantity changes in response to changes in another related quantity
- Types of infinity and limits involving infinite quantities or areas
The document defines functions, limits, derivatives, and infinity, and provides examples to illustrate these core calculus topics. It lays the groundwork for further calculus concepts to be covered like integrals, derivatives of more complex functions, and applications of limits, derivatives, and infinity.
The document discusses the Fundamental Theorem of Calculus, which has two parts. Part 1 establishes the relationship between differentiation and integration, showing that the derivative of an antiderivative is the integrand. Part 2 allows evaluation of a definite integral by evaluating the antiderivative at the bounds. Examples are given of using both parts to evaluate definite integrals. The theorem unified differentiation and integration and was fundamental to the development of calculus.
The document discusses derivatives and their applications. It begins by introducing derivatives and defining them as the rate of change of a function near an input value. It then discusses rules for finding derivatives such as the constant multiple rule, sum and difference rules, product rule, and quotient rule. Examples are given to illustrate applying these rules. The document also covers composite functions, inverse functions, second derivatives, and applications of derivatives in physics for problems involving velocity and acceleration.
This document outlines key concepts from a chapter on limits and continuity in calculus. It begins with an intuitive introduction to the limit process, providing examples to illustrate one-sided limits, limits that do not exist, and limits approaching infinity. It then defines the formal epsilon-delta definition of a limit. The document continues with sections on properties of limits, continuity, and theorems related to limits and continuity, such as the intermediate value theorem.
This document outlines key concepts from a chapter on limits and continuity in calculus. It begins with an intuitive introduction to the limit process, providing examples to illustrate one-sided limits, limits that do not exist, and limits approaching infinity. It then defines the formal epsilon-delta definition of a limit. The document continues with sections on properties of limits, continuity, and theorems related to limits and continuity, such as the intermediate value theorem.
This document outlines key concepts from a chapter on limits and continuity in calculus. It begins with an intuitive introduction to limits, including examples of how limits are used to define concepts like slope, area, and length. It discusses one-sided limits and limits that do not exist. The document then provides the formal epsilon-delta definition of a limit and discusses properties of limits. It covers continuity, types of discontinuity, and theorems related to limits and continuity. Finally, it discusses specific limits involving trigonometric and other functions.
This document outlines key concepts from a chapter on limits and continuity in calculus. It begins with an intuitive introduction to the limit process, providing examples to illustrate one-sided limits, limits that do not exist, and limits approaching infinity. It then defines the formal epsilon-delta definition of a limit. The document continues with sections on properties of limits, continuity, and theorems related to limits and continuity, such as the intermediate value theorem.
The document discusses properties and rules for evaluating limits, including:
1) The limit of a constant times a function is the constant times the limit of the function.
2) The limit of the sum/difference/product of two functions is the sum/difference/product of their individual limits.
3) One-sided limits approach from the left or right, while two-sided limits come from both sides.
4) Limits involving infinity relate to horizontal and vertical asymptotes based on the highest powers in the numerator and denominator.
5) Continuity requires the limit of a function and its value at a point to be equal.
1. The document discusses continuity of functions, including defining a continuous function as one whose graph can be traced without lifting the pencil, and defining the three conditions for a function f(x) to be continuous at a point a: f(a) must be defined, the limit of f(x) as x approaches a must exist, and the limit must equal f(a).
2. It covers types of discontinuities such as removable, jump, and infinite discontinuities.
3. Theorems are presented stating that arithmetic operations (addition, subtraction, multiplication, division) of continuous functions yield a continuous result.
4. Elementary functions like polynomials, rational fractions, and trigon
The document discusses functions and evaluating functions. It provides examples of determining if a given equation is a function using the vertical line test and evaluating functions by substituting values into the function equation. It also includes examples of evaluating composite functions using flow diagrams to illustrate the steps of evaluating each individual function.
This document discusses concepts related to calculus including limits, continuity, and derivatives of functions. Specifically, it covers:
- Definitions and theorems related to limits, continuity, and derivatives of algebraic functions.
- Evaluating limits, determining continuity of functions, and taking derivatives of algebraic functions using basic theorems of differentiation.
- The objective is for students to be able to evaluate limits, determine continuity, and find derivatives of continuous algebraic functions in explicit or implicit form after discussing these calculus concepts.
Derivatives and it’s simple applicationsRutuja Gholap
The document provides an introduction to derivatives and their applications. It defines the derivative as the rate of change of a function near an input value and discusses how it relates geometrically to the slope of the tangent line. It then gives examples of finding the derivatives of common functions like constants, polynomials, and exponentials. The document also covers basic derivative rules like the constant multiple rule, sum and difference rules, product rule, and quotient rule. Finally, it discusses applications of derivatives in topics like physics, such as calculating velocity and acceleration from a position function.
AIOU Code 803 Mathematics for Economists Semester Spring 2022 Assignment 2.pptxZawarali786
Skilling Foundation
Download Free
Past Papers
Guess Papers
Solved Assignments
Solved Thesis
Solved Lesson Plans
PDF Books
Skilling.pk
Other Websites
Diya.pk
Stamflay.com
Please Subscribe Our YouTube Channel
Skilling Foundation:https://bit.ly/3kEJI0q
WordPress Tutorials:https://bit.ly/3rqcgfE
Stamflay:https://bit.ly/2AoClW8
Please Contact at:
0314-4646739
0332-4646739
0336-4646739
اگر آپ تعلیمی نیوز، رجسٹریشن، داخلہ، ڈیٹ شیٹ، رزلٹ، اسائنمنٹ،جابز اور باقی تمام اپ ڈیٹس اپنے موبائل پر فری حاصل کرنا چاہتے ہیں ۔تو نیچے دیے گئے واٹس ایپ نمبرکو اپنے موبائل میں سیو کرکے اپنا نام لکھ کر واٹس ایپ کر دیں۔ سٹیٹس روزانہ لازمی چیک کریں۔
نوٹ : اس کے علاوہ تمام یونیورسٹیز کے آن لائن داخلے بھجوانے اور جابز کے لیے آن لائن اپلائی کروانے کے لیے رابطہ کریں۔
The document discusses rational functions and their graphs. It defines rational functions as the ratio of two polynomial functions. Key aspects covered include:
- The domain of a rational function
- Transformations of the reciprocal function 1/x and how they affect its graph
- Vertical and horizontal asymptotes of rational functions
- End behavior, intercepts, and characteristics of graphs of general rational functions
- Examples of finding asymptotes and graphing specific rational functions
The document discusses various types of integrals and rules for finding antiderivatives. It defines definite and indefinite integrals. It then lists and explains the main antiderivative rules for powers, chain rule, product rule, quotient rule, scalar multiples, sums and differences, trigonometric functions, and inverse trigonometric functions. Examples are provided to illustrate each rule.
This document contains a summary of key concepts in mathematics functions including:
- Definitions of different types of functions such as polynomial, rational, trigonometric, exponential, and logarithmic functions.
- Concepts related to functions like domain, range, and notation for functions.
- Examples demonstrating how to find the domain and range of functions.
- Key topics in trigonometry including trigonometric functions, trigonometric tables, addition formulas, laws of sines and cosines.
- Concepts of limits, average and instantaneous rates of change, techniques for finding limits, and limits involving infinity.
This document provides an overview of partial derivatives, which are used to analyze functions with multiple variables. Key topics covered include:
- Definitions of limits, continuity, and partial derivatives for multivariable functions.
- Directional derivatives and the gradient, which describe the rate of change in a specified direction.
- The chain rule for partial derivatives, and implicit differentiation.
- Linearization and Taylor series approximations for multivariable functions.
- Finding local extrema and optimizing functions, using techniques like classifying critical points.
Integral Calculus. - Differential Calculus - Integration as an Inverse Process of Differentiation - Methods of Integration - Integration using trigonometric identities - Integrals of Some Particular Functions - rational function - partial fraction - Integration by partial fractions - standard integrals - First and second fundamental theorem of integral calculus
The document discusses limits and how they are used to calculate the derivative of a function. It defines what it means for a sequence to approach a limit from the right or left side. Graphs and examples are provided to illustrate these concepts. The key rules for calculating limits are outlined, such as using algebra to split limits into their constituent parts. Common types of obvious limits are also stated, such as limits of constants or products involving constants.
The document discusses the concept of limits and clarifies the notation used to describe sequences approaching a number. It explains that saying "x approaches 0 from the right side" means the sequence values only become smaller than 0 after a finite number of terms. Similarly, approaching from the left means only finitely many terms are greater than 0. The direction a sequence approaches a number affects limits like the limit of |x|/x as x approaches 0.
This document discusses two sections, Section 3.1 and Section 3.3, but provides no details about the content or topics covered in either section. The document gives the section numbers and titles but no other informative or descriptive text.
The document discusses calculating the area of a region R. It introduces using a ruler x to measure the span of R from x=a to x=b. It defines the cross-sectional length L(x) and partitions the interval [a,b] into subintervals. The Riemann sum of the areas of approximating rectangles is shown to approach the actual area of R, defined as the definite integral of L(x) from a to b. As an example, it calculates the area between the curves y=-x^2+2x and y=x^2 by finding the interval spans from 0 to 1 and taking the integral of the difference of the functions.
The document summarizes different types of derivatives:
Simple derivatives involve a single input and output. Implicit derivatives are taken for equations with two or more variables, treating one as the independent variable. An example finds derivatives of u with respect to v and v with respect to u for the equation 2u^2 - v^3 = 2 - uv. The derivatives are related by the reciprocal relationship in differential notation.
This document contains 20 math word problems involving rates of change of quantities like distance, area, radius, and volume over time. The problems involve concepts like expanding derivatives, rectangles changing size, cars moving at intersections, distances between moving objects, water filling and draining from tanks, ladders on houses, waves expanding in water, balloons deflating, and water filling triangular troughs. Rates of change are calculated for variables like length, width, area, distance, radius, and volume at specific values over time.
The document contains 10 multi-part exercises involving calculating rates of change, finding maximums and optima, and approximating changes in functions. The exercises involve concepts like linear price-demand functions, surface area and volume relationships for geometric objects, and force functions related to physics concepts like gravity and electric force.
1. The document provides instructions for using calculus concepts like derivatives and integrals to approximate values. It contains 14 problems involving finding derivatives, using derivatives to approximate values, finding volumes with integrals, and using Newton's method to find roots of functions.
2. The final problem asks to use Newton's method in Excel to find the two roots of the function y = ex - 2x - 2 that exist between -3 and 3 to 5 decimal places, and then justify that the approximations are correct.
This document contains 16 multi-part math problems involving optimization of functions, geometry, and physics. The problems cover topics like finding extrema of functions, finding points on lines, maximizing areas of geometric shapes given constraints, minimizing materials needed to construct cylinders and fences, and finding positions of maximum or minimum values of physical quantities like force and illumination.
The document discusses how derivatives can represent rates of change. It states that given a function f(x), the derivative f'(a) is equivalent to the slope of the tangent line at x=a, the instantaneous rate of change of y with respect to x at x=a, and the amount of change in y for a 1 unit change in x at x=a. It then provides an example using a price-demand function for chickens, finding that the maximum revenue of $1152 occurs at a price of $10 per chicken.
1. Graph and analyze the critical points, extrema, inflection points, intervals of increasing/decreasing, and intervals of concave up/down for 10 functions.
2. Review homework on finding derivatives using the definition of the difference quotient and evaluating limits. Find the derivatives of 6 functions.
3. Use implicit differentiation to find the derivative of one function defined implicitly and to find points with tangent lines of slope 1 for another implicit function.
4. Find the second derivatives of two functions.
1) The document provides a tutorial on using formulas in Excel, including how to enter formulas, use relative and absolute cell references, perform calculations on ranges of cells, and sum columns of data.
2) It includes steps to enter sample data, calculate values like x-squared and frequencies multiplied by x and x-squared, and use formulas to automatically calculate those values down a column.
3) The tutorial concludes with instructions to sum the sample data columns, enter the student's name, save the Excel file, and provide a printout.
[OReilly Superstream] Occupy the Space: A grassroots guide to engineering (an...Jason Yip
The typical problem in product engineering is not bad strategy, so much as “no strategy”. This leads to confusion, lack of motivation, and incoherent action. The next time you look for a strategy and find an empty space, instead of waiting for it to be filled, I will show you how to fill it in yourself. If you’re wrong, it forces a correction. If you’re right, it helps create focus. I’ll share how I’ve approached this in the past, both what works and lessons for what didn’t work so well.
QR Secure: A Hybrid Approach Using Machine Learning and Security Validation F...AlexanderRichford
QR Secure: A Hybrid Approach Using Machine Learning and Security Validation Functions to Prevent Interaction with Malicious QR Codes.
Aim of the Study: The goal of this research was to develop a robust hybrid approach for identifying malicious and insecure URLs derived from QR codes, ensuring safe interactions.
This is achieved through:
Machine Learning Model: Predicts the likelihood of a URL being malicious.
Security Validation Functions: Ensures the derived URL has a valid certificate and proper URL format.
This innovative blend of technology aims to enhance cybersecurity measures and protect users from potential threats hidden within QR codes 🖥 🔒
This study was my first introduction to using ML which has shown me the immense potential of ML in creating more secure digital environments!
MySQL InnoDB Storage Engine: Deep Dive - MydbopsMydbops
This presentation, titled "MySQL - InnoDB" and delivered by Mayank Prasad at the Mydbops Open Source Database Meetup 16 on June 8th, 2024, covers dynamic configuration of REDO logs and instant ADD/DROP columns in InnoDB.
This presentation dives deep into the world of InnoDB, exploring two ground-breaking features introduced in MySQL 8.0:
• Dynamic Configuration of REDO Logs: Enhance your database's performance and flexibility with on-the-fly adjustments to REDO log capacity. Unleash the power of the snake metaphor to visualize how InnoDB manages REDO log files.
• Instant ADD/DROP Columns: Say goodbye to costly table rebuilds! This presentation unveils how InnoDB now enables seamless addition and removal of columns without compromising data integrity or incurring downtime.
Key Learnings:
• Grasp the concept of REDO logs and their significance in InnoDB's transaction management.
• Discover the advantages of dynamic REDO log configuration and how to leverage it for optimal performance.
• Understand the inner workings of instant ADD/DROP columns and their impact on database operations.
• Gain valuable insights into the row versioning mechanism that empowers instant column modifications.
Session 1 - Intro to Robotic Process Automation.pdfUiPathCommunity
👉 Check out our full 'Africa Series - Automation Student Developers (EN)' page to register for the full program:
https://bit.ly/Automation_Student_Kickstart
In this session, we shall introduce you to the world of automation, the UiPath Platform, and guide you on how to install and setup UiPath Studio on your Windows PC.
📕 Detailed agenda:
What is RPA? Benefits of RPA?
RPA Applications
The UiPath End-to-End Automation Platform
UiPath Studio CE Installation and Setup
💻 Extra training through UiPath Academy:
Introduction to Automation
UiPath Business Automation Platform
Explore automation development with UiPath Studio
👉 Register here for our upcoming Session 2 on June 20: Introduction to UiPath Studio Fundamentals: https://community.uipath.com/events/details/uipath-lagos-presents-session-2-introduction-to-uipath-studio-fundamentals/
The Department of Veteran Affairs (VA) invited Taylor Paschal, Knowledge & Information Management Consultant at Enterprise Knowledge, to speak at a Knowledge Management Lunch and Learn hosted on June 12, 2024. All Office of Administration staff were invited to attend and received professional development credit for participating in the voluntary event.
The objectives of the Lunch and Learn presentation were to:
- Review what KM ‘is’ and ‘isn’t’
- Understand the value of KM and the benefits of engaging
- Define and reflect on your “what’s in it for me?”
- Share actionable ways you can participate in Knowledge - - Capture & Transfer
"What does it really mean for your system to be available, or how to define w...Fwdays
We will talk about system monitoring from a few different angles. We will start by covering the basics, then discuss SLOs, how to define them, and why understanding the business well is crucial for success in this exercise.
ScyllaDB is making a major architecture shift. We’re moving from vNode replication to tablets – fragments of tables that are distributed independently, enabling dynamic data distribution and extreme elasticity. In this keynote, ScyllaDB co-founder and CTO Avi Kivity explains the reason for this shift, provides a look at the implementation and roadmap, and shares how this shift benefits ScyllaDB users.
"Scaling RAG Applications to serve millions of users", Kevin GoedeckeFwdays
How we managed to grow and scale a RAG application from zero to thousands of users in 7 months. Lessons from technical challenges around managing high load for LLMs, RAGs and Vector databases.
How information systems are built or acquired puts information, which is what they should be about, in a secondary place. Our language adapted accordingly, and we no longer talk about information systems but applications. Applications evolved in a way to break data into diverse fragments, tightly coupled with applications and expensive to integrate. The result is technical debt, which is re-paid by taking even bigger "loans", resulting in an ever-increasing technical debt. Software engineering and procurement practices work in sync with market forces to maintain this trend. This talk demonstrates how natural this situation is. The question is: can something be done to reverse the trend?
Northern Engraving | Nameplate Manufacturing Process - 2024Northern Engraving
Manufacturing custom quality metal nameplates and badges involves several standard operations. Processes include sheet prep, lithography, screening, coating, punch press and inspection. All decoration is completed in the flat sheet with adhesive and tooling operations following. The possibilities for creating unique durable nameplates are endless. How will you create your brand identity? We can help!
inQuba Webinar Mastering Customer Journey Management with Dr Graham HillLizaNolte
HERE IS YOUR WEBINAR CONTENT! 'Mastering Customer Journey Management with Dr. Graham Hill'. We hope you find the webinar recording both insightful and enjoyable.
In this webinar, we explored essential aspects of Customer Journey Management and personalization. Here’s a summary of the key insights and topics discussed:
Key Takeaways:
Understanding the Customer Journey: Dr. Hill emphasized the importance of mapping and understanding the complete customer journey to identify touchpoints and opportunities for improvement.
Personalization Strategies: We discussed how to leverage data and insights to create personalized experiences that resonate with customers.
Technology Integration: Insights were shared on how inQuba’s advanced technology can streamline customer interactions and drive operational efficiency.
As AI technology is pushing into IT I was wondering myself, as an “infrastructure container kubernetes guy”, how get this fancy AI technology get managed from an infrastructure operational view? Is it possible to apply our lovely cloud native principals as well? What benefit’s both technologies could bring to each other?
Let me take this questions and provide you a short journey through existing deployment models and use cases for AI software. On practical examples, we discuss what cloud/on-premise strategy we may need for applying it to our own infrastructure to get it to work from an enterprise perspective. I want to give an overview about infrastructure requirements and technologies, what could be beneficial or limiting your AI use cases in an enterprise environment. An interactive Demo will give you some insides, what approaches I got already working for real.
Keywords: AI, Containeres, Kubernetes, Cloud Native
Event Link: https://meine.doag.org/events/cloudland/2024/agenda/#agendaId.4211
Must Know Postgres Extension for DBA and Developer during MigrationMydbops
Mydbops Opensource Database Meetup 16
Topic: Must-Know PostgreSQL Extensions for Developers and DBAs During Migration
Speaker: Deepak Mahto, Founder of DataCloudGaze Consulting
Date & Time: 8th June | 10 AM - 1 PM IST
Venue: Bangalore International Centre, Bangalore
Abstract: Discover how PostgreSQL extensions can be your secret weapon! This talk explores how key extensions enhance database capabilities and streamline the migration process for users moving from other relational databases like Oracle.
Key Takeaways:
* Learn about crucial extensions like oracle_fdw, pgtt, and pg_audit that ease migration complexities.
* Gain valuable strategies for implementing these extensions in PostgreSQL to achieve license freedom.
* Discover how these key extensions can empower both developers and DBAs during the migration process.
* Don't miss this chance to gain practical knowledge from an industry expert and stay updated on the latest open-source database trends.
Mydbops Managed Services specializes in taking the pain out of database management while optimizing performance. Since 2015, we have been providing top-notch support and assistance for the top three open-source databases: MySQL, MongoDB, and PostgreSQL.
Our team offers a wide range of services, including assistance, support, consulting, 24/7 operations, and expertise in all relevant technologies. We help organizations improve their database's performance, scalability, efficiency, and availability.
Contact us: info@mydbops.com
Visit: https://www.mydbops.com/
Follow us on LinkedIn: https://in.linkedin.com/company/mydbops
For more details and updates, please follow up the below links.
Meetup Page : https://www.meetup.com/mydbops-databa...
Twitter: https://twitter.com/mydbopsofficial
Blogs: https://www.mydbops.com/blog/
Facebook(Meta): https://www.facebook.com/mydbops/
2. Computations of Derivatives
The algebra for computing derivatives using the limit
approach gets unmanageable fast.
3. Computations of Derivatives
The algebra for computing derivatives using the limit
approach gets unmanageable fast.
Elementary functions are built in finitely many steps
using a few basic formulas, the real numbers and
the algebraic operations +,–, *, / and function
composition
4. Computations of Derivatives
The algebra for computing derivatives using the limit
approach gets unmanageable fast.
Elementary functions are built in finitely many steps
using a few basic formulas, the real numbers and
the algebraic operations +,–, *, / and function
composition–plug in a formula into another formula.
5. Computations of Derivatives
The algebra for computing derivatives using the limit
approach gets unmanageable fast.
Elementary functions are built in finitely many steps
using a few basic formulas, the real numbers and
the algebraic operations +,–, *, / and function
composition–plug in a formula into another formula.
It’s because of their structures that elementary
functions are the ones we are able to find derivatives
easily.
6. Computations of Derivatives
The algebra for computing derivatives using the limit
approach gets unmanageable fast.
Elementary functions are built in finitely many steps
using a few basic formulas, the real numbers and
the algebraic operations +,–, *, / and function
composition–plug in a formula into another formula.
It’s because of their structures that elementary
functions are the ones we are able to find derivatives
easily.
The operation of taking derivatives is called
differentiation.
7. Computations of Derivatives
The algebra for computing derivatives using the limit
approach gets unmanageable fast.
Elementary functions are built in finitely many steps
using a few basic formulas, the real numbers and
the algebraic operations +,–, *, / and function
composition–plug in a formula into another formula.
It’s because of their structures that elementary
functions are the ones we are able to find derivatives
easily.
The operation of taking derivatives is called
differentiation. We will examine how the operation of
differentiation behaves under the above operations.
8. Computations of Derivatives
The following properties of limits pass on directly to
the differentiation operation.
9. Computations of Derivatives
The following properties of limits pass on directly to
the differentiation operation. These are the sum,
difference, and constant multiplications rules of limits:
lim (f ± g) = lim f ± lim g and that lim c(f) = c lim f
10. Computations of Derivatives
The following properties of limits pass on directly to
the differentiation operation. These are the sum,
difference, and constant multiplications rules of limits:
lim (f ± g) = lim f ± lim g and that lim c(f) = c lim f
It’s important to point out here that the corresponding
product and quotient properties of limits do not pass
on to the differentiation operation.
11. Computations of Derivatives
The following properties of limits pass on directly to
the differentiation operation. These are the sum,
difference, and constant multiplications rules of limits:
lim (f ± g) = lim f ± lim g and that lim c(f) = c lim f
It’s important to point out here that the corresponding
product and quotient properties of limits do not pass
on to the differentiation operation. The product and
quotient rules of differentiation are more complicated.
12. Computations of Derivatives
The following properties of limits pass on directly to
the differentiation operation. These are the sum,
difference, and constant multiplications rules of limits:
lim (f ± g) = lim f ± lim g and that lim c(f) = c lim f
It’s important to point out here that the corresponding
product and quotient properties of limits do not pass
on to the differentiation operation. The product and
quotient rules of differentiation are more complicated.
Unless stated otherwise, we assume that the
derivatives at x exist in all the theorems below.
13. Computations of Derivatives
The following properties of limits pass on directly to
the differentiation operation. These are the sum,
difference, and constant multiplications rules of limits:
lim (f ± g) = lim f ± lim g and that lim c(f) = c lim f
It’s important to point out here that the corresponding
product and quotient properties of limits do not pass
on to the differentiation operation. The product and
quotient rules of differentiation are more complicated.
Unless stated otherwise, we assume that the
derivatives at x exist in all the theorems below.
The ± and Constant–Multiple Derivative Rules
14. Computations of Derivatives
The following properties of limits pass on directly to
the differentiation operation. These are the sum,
difference, and constant multiplications rules of limits:
lim (f ± g) = lim f ± lim g and that lim c(f) = c lim f
It’s important to point out here that the corresponding
product and quotient properties of limits do not pass
on to the differentiation operation. The product and
quotient rules of differentiation are more complicated.
Unless stated otherwise, we assume that the
derivatives at x exist in all the theorems below.
The ± and Constant–Multiple Derivative Rules
Let f(x) and g(x) be two functions then
15. Computations of Derivatives
The following properties of limits pass on directly to
the differentiation operation. These are the sum,
difference, and constant multiplications rules of limits:
lim (f ± g) = lim f ± lim g and that lim c(f) = c lim f
It’s important to point out here that the corresponding
product and quotient properties of limits do not pass
on to the differentiation operation. The product and
quotient rules of differentiation are more complicated.
Unless stated otherwise, we assume that the
derivatives at x exist in all the theorems below.
The ± and Constant–Multiple Derivative Rules
Let f(x) and g(x) be two functions then
i. (f(x)±g(x)) ' = f '(x)±g '(x)
ii. (cf(x)) ' = c*f '(x) where c is a constant.
17. Computations of Derivatives
To verify i, note that
lim
[f(x+h) + g(x+h)] – [f(x) + g(x)]
h →0 h
=
(f(x) + g(x)) '
18. Computations of Derivatives
To verify i, note that
lim
[f(x+h) + g(x+h)] – [f(x) + g(x)]
h →0 h
=
(f(x) + g(x)) '
lim [f(x+h) – f(x)] + [g(x+h) – g(x)]
h h →0
=
19. Computations of Derivatives
To verify i, note that
lim
[f(x+h) + g(x+h)] – [f(x) + g(x)]
h →0 h
=
(f(x) + g(x)) '
lim [f(x+h) – f(x)] + [g(x+h) – g(x)]
h h →0
=
lim [f(x+h) – f(x)] [g(x+h) – g(x)]
= +
h
h →0
h
20. Computations of Derivatives
To verify i, note that
lim
[f(x+h) + g(x+h)] – [f(x) + g(x)]
h →0 h
=
(f(x) + g(x)) '
lim [f(x+h) – f(x)] + [g(x+h) – g(x)]
h h →0
=
lim [f(x+h) – f(x)] [g(x+h) – g(x)]
= +
h
h →0
h
(the sum property of limits)
lim [f(x+h) – f(x)]
h h →0 = +
lim [g(x+h) – g(x)]
h →0
h
21. Computations of Derivatives
To verify i, note that
lim
[f(x+h) + g(x+h)] – [f(x) + g(x)]
h →0 h
=
(f(x) + g(x)) '
lim [f(x+h) – f(x)] + [g(x+h) – g(x)]
h h →0
=
lim [f(x+h) – f(x)] [g(x+h) – g(x)]
= +
h
h →0
h
(the sum property of limits)
lim [f(x+h) – f(x)]
h h →0 = +
lim [g(x+h) – g(x)]
h →0
h
= f '(x) + g '(x)
22. Computations of Derivatives
To verify i, note that
lim
[f(x+h) + g(x+h)] – [f(x) + g(x)]
h →0 h
=
(f(x) + g(x)) '
lim [f(x+h) – f(x)] + [g(x+h) – g(x)]
h h →0
=
lim [f(x+h) – f(x)] [g(x+h) – g(x)]
= +
h
h →0
h
(the sum property of limits)
lim [f(x+h) – f(x)]
h h →0 = +
lim [g(x+h) – g(x)]
h →0
h
= f '(x) + g '(x)
Your turn: Verity part ii in a similar manner.
23. Computations of Derivatives
Geometrically, (f + g) ' = f ' + g ' says that slope of
the sum function is the sum of the slopes at any point.
24. Computations of Derivatives
Geometrically, (f + g) ' = f ' + g ' says that slope of
the sum function is the sum of the slopes at any point.
Let’s illustrate this with graphs.
25. Computations of Derivatives
Geometrically, (f + g) ' = f ' + g ' says that slope of
the sum function is the sum of the slopes at any point.
Let’s illustrate this with graphs.
Let the slope at x = a of f(x) be 1/3
or f '(a) = 1/3 and that
the slope at x = a of g(x) is 2/3
or g '(a) = 2/3,
26. Computations of Derivatives
Geometrically, (f + g) ' = f ' + g ' says that slope of
the sum function is the sum of the slopes at any point.
Let’s illustrate this with graphs.
Let the slope at x = a of f(x) be 1/3
or f '(a) = 1/3 and that
the slope at x = a of g(x) is 2/3
or g '(a) = 2/3,
slope = 2/3
slope = 1/3
27. Computations of Derivatives
Geometrically, (f + g) ' = f ' + g ' says that slope of
the sum function is the sum of the slopes at any point.
Let’s illustrate this with graphs.
Let the slope at x = a of f(x) be 1/3
or f '(a) = 1/3 and that
the slope at x = a of g(x) is 2/3
or g '(a) = 2/3,
then the slope at x = a of (f + g)(x)
slope = 2/3
slope = 1/3
28. Computations of Derivatives
Geometrically, (f + g) ' = f ' + g ' says that slope of
the sum function is the sum of the slopes at any point.
Let’s illustrate this with graphs.
y=(f + g)(x)
Let the slope at x = a of f(x) be 1/3
or f '(a) = 1/3 and that
the slope at x = a of g(x) is 2/3
or g '(a) = 2/3,
then the slope at x = a of (f + g)(x)
slope = 2/3
slope = 1/3
29. Computations of Derivatives
Geometrically, (f + g) ' = f ' + g ' says that slope of
the sum function is the sum of the slopes at any point.
Let’s illustrate this with graphs.
y=(f + g)(x)
Let the slope at x = a of f(x) be 1/3
or f '(a) = 1/3 and that
the slope at x = a of g(x) is 2/3
or g '(a) = 2/3,
then the slope at x = a of (f + g)(x)
slope = 2/3
slope = 1/3
30. Computations of Derivatives
Geometrically, (f + g) ' = f ' + g ' says that slope of
the sum function is the sum of the slopes at any point.
Let’s illustrate this with graphs.
y=(f + g)(x)
Let the slope at x = a of f(x) be 1/3
or f '(a) = 1/3 and that
the slope at x = a of g(x) is 2/3
or g '(a) = 2/3,
then the slope at x = a of (f + g)(x)
or (f + g)'(a) = 1/3 + 2/3 = 1
slope
=1/3+2/3
= 1
slope = 2/3
slope = 1/3
31. Computations of Derivatives
Geometrically, (f + g) ' = f ' + g ' says that slope of
the sum function is the sum of the slopes at any point.
Let’s illustrate this with graphs.
y=(f + g)(x)
Let the slope at x = a of f(x) be 1/3
or f '(a) = 1/3 and that
the slope at x = a of g(x) is 2/3
or g '(a) = 2/3,
then the slope at x = a of (f + g)(x)
or (f + g)'(a) = 1/3 + 2/3 = 1
slope
=1/3+2/3
= 1
slope = 2/3
slope = 1/3
Likewise c*f(x) have the slope
c*f '(x) as the slope at x = a.
32. Computations of Derivatives
Geometrically, (f + g) ' = f ' + g ' says that slope of
the sum function is the sum of the slopes at any point.
Let’s illustrate this with graphs.
y=(f + g)(x)
Let the slope at x = a of f(x) be 1/3
or f '(a) = 1/3 and that
the slope at x = a of g(x) is 2/3
or g '(a) = 2/3,
then the slope at x = a of (f + g)(x)
or (f + g)'(a) = 1/3 + 2/3 = 1
slope
=1/3+2/3
= 1
slope = 2/3
Likewise c*f(x) have the slope
c*f '(x) as the slope at x = a.
The slope of y = 2x – 1 is 2,
the slope of 3(2x – 1) = 6x – 3 is 6.
slope = 1/3
33. Computations of Derivatives
However the product or quotient rule of limits that
f
lim f
lim (f)(g) = (lim f)*(lim g) and lim
g =
lim g
bear no direct relation for computing derivatives.
34. Computations of Derivatives
However the product or quotient rule of limits that
f
lim f
lim (f)(g) = (lim f)*(lim g) and lim
g =
lim g
bear no direct relation for computing derivatives.
( ) f '
fg
!! (f * g)' ≠ f ' * g ' ≠ g '
35. Computations of Derivatives
However the product or quotient rule of limits that
f
lim f
lim (f)(g) = (lim f)*(lim g) and lim
g =
lim g
bear no direct relation for computing derivatives.
fg
!! (f * g)' ≠ f ' * g ' ( )
≠ g '
f '
The Product and Quotient Rules of Derivatives
36. Computations of Derivatives
However the product or quotient rule of limits that
f
lim f
lim (f)(g) = (lim f)*(lim g) and lim
g =
lim g
bear no direct relation for computing derivatives.
fg
!! (f * g)' ≠ f ' * g ' ( )
≠
f '
g '
The Product and Quotient Rules of Derivatives
Write f for f(x) and g for g(x) then
(fg)' = f'g + fg'
37. Computations of Derivatives
However the product or quotient rule of limits that
f
lim f
lim (f)(g) = (lim f)*(lim g) and lim
g =
lim g
bear no direct relation for computing derivatives.
fg
!! (f * g)' ≠ f ' * g ' ( )
≠
f '
g '
The Product and Quotient Rules of Derivatives
Write f for f(x) and g for g(x) then
(fg)' = f'g + fg'
gf ' – fg'
g2
f
( g)' =
38. Computations of Derivatives
However the product or quotient rule of limits that
f
lim f
lim (f)(g) = (lim f)*(lim g) and lim
g =
lim g
bear no direct relation for computing derivatives.
fg
!! (f * g)' ≠ f ' * g ' ( )
≠
f '
g '
The Product and Quotient Rules of Derivatives
Write f for f(x) and g for g(x) then
(fg)' = f'g + fg'
f
=
gf ' – fg'
( g)' g2
Here are sites for the verifications of these rules.
http://en.wikipedia.org/wiki/Product_rule#Proof_of_the_product_rule
http://en.wikipedia.org/wiki/Quotient_rule
40. Computations of Derivatives
Here is a geometric analogy of the Product Rule.
Given a rectangle of F * G,
F
G
41. Computations of Derivatives
Here is a geometric analogy of the Product Rule.
Given a rectangle of F * G, Suppose F is extended to
F + ΔF and G is extended to G + ΔG
F
G
42. Computations of Derivatives
Here is a geometric analogy of the Product Rule.
Given a rectangle of F * G, Suppose F is extended to
F + ΔF and G is extended to G + ΔG
F
G
ΔF
ΔG
43. Computations of Derivatives
Here is a geometric analogy of the Product Rule.
Given a rectangle of F * G, Suppose F is extended to
F + ΔF and G is extended to G + ΔG
F
G
ΔF
ΔG
The difference in the two
areas is
(F+ΔF)(G+ΔG) – FG
44. Computations of Derivatives
Here is a geometric analogy of the Product Rule.
Given a rectangle of F * G, Suppose F is extended to
F + ΔF and G is extended to G + ΔG
F
G
ΔF
ΔG
The difference in the two
areas is
(F+ΔF)(G+ΔG) – FG
= shaded area
45. Computations of Derivatives
Here is a geometric analogy of the Product Rule.
Given a rectangle of F * G, Suppose F is extended to
F + ΔF and G is extended to G + ΔG
F
G
ΔF
ΔG
The difference in the two
areas is
(F+ΔF)(G+ΔG) – FG
= shaded area
= three parts
46. Computations of Derivatives
Here is a geometric analogy of the Product Rule.
Given a rectangle of F * G, Suppose F is extended to
F + ΔF and G is extended to G + ΔG
F
G
ΔF
ΔG
The difference in the two
areas is
(F+ΔF)(G+ΔG) – FG
= shaded area
= three parts
= F*ΔG F*ΔG
47. Computations of Derivatives
Here is a geometric analogy of the Product Rule.
Given a rectangle of F * G, Suppose F is extended to
F + ΔF and G is extended to G + ΔG
F
G
ΔF
ΔG
The difference in the two
areas is
ΔF*G
(F+ΔF)(G+ΔG) – FG
= shaded area
= three parts
= F*ΔG+ΔF*G F*ΔG
48. Computations of Derivatives
Here is a geometric analogy of the Product Rule.
Given a rectangle of F * G, Suppose F is extended to
F + ΔF and G is extended to G + ΔG
F
G
ΔF
ΔG
The difference in the two
areas is
ΔF*G
(F+ΔF)(G+ΔG) – FG
= shaded area
= three parts
= F*ΔG+ΔF*G+ΔF*ΔG. F*ΔG
ΔF*ΔG
49. Computations of Derivatives
Here is a geometric analogy of the Product Rule.
Given a rectangle of F * G, Suppose F is extended to
F + ΔF and G is extended to G + ΔG
F
G
ΔF
ΔG
The difference in the two
areas is
ΔF*G
(F+ΔF)(G+ΔG) – FG
= shaded area
= three parts
= F*ΔG+ΔF*G+ΔF*ΔG. F*ΔG
ΔF*ΔG
If ΔF and ΔG are small then ΔF*ΔG area is negligible
50. Computations of Derivatives
Here is a geometric analogy of the Product Rule.
Given a rectangle of F * G, Suppose F is extended to
F + ΔF and G is extended to G + ΔG
F
G
ΔF
ΔG
The difference in the two
areas is
ΔF*G
(F+ΔF)(G+ΔG) – FG
= shaded area
= three parts
= F*ΔG+ΔF*G+ΔF*ΔG. F*ΔG
ΔF*ΔG
If ΔF and ΔG are small then ΔF*ΔG area is negligible
so(F+ΔF)(G+ΔG) – FG ≈ F*ΔG+ΔF*G
51. Computations of Derivatives
Knowing how the differentiation operation ( ) '
interacts with the algebraic operations +, –, *, and /,
enables us develop the algebra to take the derivatives
of polynomials and rational functions.
52. Computations of Derivatives
Knowing how the differentiation operation ( ) '
interacts with the algebraic operations +, –, *, and /,
enables us develop the algebra to take the derivatives
of polynomials and rational functions.
Derivatives of Polynomials and Rational Functions
53. Computations of Derivatives
Knowing how the differentiation operation ( ) '
interacts with the algebraic operations +, –, *, and /,
enables us develop the algebra to take the derivatives
of polynomials and rational functions.
Derivatives of Polynomials and Rational Functions
We start with the observations that
i. If f(x) = c, a constant function, then f '(x) = (c) ' = 0.
54. Computations of Derivatives
Knowing how the differentiation operation ( ) '
interacts with the algebraic operations +, –, *, and /,
enables us develop the algebra to take the derivatives
of polynomials and rational functions.
Derivatives of Polynomials and Rational Functions
We start with the observations that
i. If f(x) = c, a constant function, then f '(x) = (c) ' = 0.
ii. If f(x) = x, the identity function, then f '(x) = (x) ' = 1.
55. Computations of Derivatives
Knowing how the differentiation operation ( ) '
interacts with the algebraic operations +, –, *, and /,
enables us develop the algebra to take the derivatives
of polynomials and rational functions.
Derivatives of Polynomials and Rational Functions
We start with the observations that
i. If f(x) = c, a constant function, then f '(x) = (c) ' = 0.
ii. If f(x) = x, the identity function, then f '(x) = (x) ' = 1.
If we view x2 as the product of f(x)*g(x) where
f(x) = g(x) = x,
56. Computations of Derivatives
Knowing how the differentiation operation ( ) '
interacts with the algebraic operations +, –, *, and /,
enables us develop the algebra to take the derivatives
of polynomials and rational functions.
Derivatives of Polynomials and Rational Functions
We start with the observations that
i. If f(x) = c, a constant function, then f '(x) = (c) ' = 0.
ii. If f(x) = x, the identity function, then f '(x) = (x) ' = 1.
If we view x2 as the product of f(x)*g(x) where
f(x) = g(x) = x, then
(x2)' = (fg)’
57. Computations of Derivatives
Knowing how the differentiation operation ( ) '
interacts with the algebraic operations +, –, *, and /,
enables us develop the algebra to take the derivatives
of polynomials and rational functions.
Derivatives of Polynomials and Rational Functions
We start with the observations that
i. If f(x) = c, a constant function, then f '(x) = (c) ' = 0.
ii. If f(x) = x, the identity function, then f '(x) = (x) ' = 1.
If we view x2 as the product of f(x)*g(x) where
f(x) = g(x) = x, then
(x2)' = (fg)' = (f)'g + f(g)' by the Product Rule
58. Computations of Derivatives
Knowing how the differentiation operation ( ) '
interacts with the algebraic operations +, –, *, and /,
enables us develop the algebra to take the derivatives
of polynomials and rational functions.
Derivatives of Polynomials and Rational Functions
We start with the observations that
i. If f(x) = c, a constant function, then f '(x) = (c) ' = 0.
ii. If f(x) = x, the identity function, then f '(x) = (x) ' = 1.
If we view x2 as the product of f(x)*g(x) where
f(x) = g(x) = x, then
(x2)' = (fg)' = (f)'g + f(g)' by the Product Rule
= (1)x + x(1) by the fact (x)' = 1
59. Computations of Derivatives
Knowing how the differentiation operation ( ) '
interacts with the algebraic operations +, –, *, and /,
enables us develop the algebra to take the derivatives
of polynomials and rational functions.
Derivatives of Polynomials and Rational Functions
We start with the observations that
i. If f(x) = c, a constant function, then f '(x) = (c) ' = 0.
ii. If f(x) = x, the identity function, then f '(x) = (x) ' = 1.
If we view x2 as the product of f(x)*g(x) where
f(x) = g(x) = x, then
(x2)' = (fg)' = (f)'g + f(g)' by the Product Rule
= (1)x + x(1) by the fact (x)' = 1
or that (x2)'= 2x
61. Computations of Derivatives
Because (x2)' = 2x, therefore the derivative of –2x2 is
(–2x2)' = –2(x2)'
We say we “pull out the
constant “ when we use the
Constant Multiple Rule.
62. Computations of Derivatives
Because (x2)' = 2x, therefore the derivative of –2x2 is
(–2x2)' = –2(x2)' = –2(2x) = –4x.
We say we “pull out the
constant “ when we use the
Constant Multiple Rule.
63. Computations of Derivatives
Because (x2)' = 2x, therefore the derivative of –2x2 is
(–2x2)' = –2(x2)' = –2(2x) = –4x.
To calculate
(–2x2 + 2x + 1)' =
64. Computations of Derivatives
Because (x2)' = 2x, therefore the derivative of –2x2 is
(–2x2)' = –2(x2)' = –2(2x) = –4x.
To calculate
“ take derivative
term by term “
(–2x2 + 2x + 1)' = (–2x2)' + (2x)' + (1)'
65. Computations of Derivatives
Because (x2)' = 2x, therefore the derivative of –2x2 is
(–2x2)' = –2(x2)' = –2(2x) = –4x.
To calculate
“ take derivative
term by term “
(–2x2 + 2x + 1)' = (–2x2)' + (2x)' + (1)'
“pull out the constant “
= –2(x2)' + 2(x)'
66. Computations of Derivatives
Because (x2)' = 2x, therefore the derivative of –2x2 is
(–2x2)' = –2(x2)' = –2(2x) = –4x.
To calculate
(–2x2 + 2x + 1)' = (–2x2)' + (2x)' + (1)'
“pull out the constant “
= –2(x2)' + 2(x)' + 0
“derivative of constant = 0“
“ take derivative
term by term “
67. Computations of Derivatives
Because (x2)' = 2x, therefore the derivative of –2x2 is
(–2x2)' = –2(x2)' = –2(2x) = –4x.
To calculate
“ take derivative
term by term “
(–2x2 + 2x + 1)' = (–2x2)' + (2x)' + (1)'
“pull out the constant “
= –2(x2)' + 2(x)' + 0
“derivative of constant = 0“
= –2(2x) + 2(1)
68. Computations of Derivatives
Because (x2)' = 2x, therefore the derivative of –2x2 is
(–2x2)' = –2(x2)' = –2(2x) = –4x.
To calculate
“ take derivative
term by term “
(–2x2 + 2x + 1)' = (–2x2)' + (2x)' + (1)'
“pull out the constant “
= –2(x2)' + 2(x)' + 0
“derivative of constant = 0“
= –2(2x) + 2(1)
= –4x + 2
69. Computations of Derivatives
Because (x2)' = 2x, therefore the derivative of –2x2 is
(–2x2)' = –2(x2)' = –2(2x) = –4x.
To calculate
“ take derivative
term by term “
(–2x2 + 2x + 1)' = (–2x2)' + (2x)' + (1)'
“pull out the constant “
= –2(x2)' + 2(x)' + 0
“derivative of constant = 0“
= –2(2x) + 2(1)
= –4x + 2
This method is a lot easier than the limit method
required by the definition.
71. Computations of Derivatives
If we view x3 as the product of f(x)*g(x) where
f(x) = x2 and g(x) = x, then
(x3)' = (fg)' = (f)'g + f(g)' by the Product Rule
72. Computations of Derivatives
If we view x3 as the product of f(x)*g(x) where
f(x) = x2 and g(x) = x, then
(x3)' = (fg)' = (f)'g + f(g)' by the Product Rule
= (x2)'x + x2(x)'
73. Computations of Derivatives
If we view x3 as the product of f(x)*g(x) where
f(x) = x2 and g(x) = x, then
(x3)' = (fg)' = (f)'g + f(g)' by the Product Rule
= (x2)'x + x2(x)' = (2x)x + x2(1)
74. Computations of Derivatives
If we view x3 as the product of f(x)*g(x) where
f(x) = x2 and g(x) = x, then
(x3)' = (fg)' = (f)'g + f(g)' by the Product Rule
= (x2)'x + x2(x)' = (2x)x + x2(1)
so (x3)'= 3x2
75. Computations of Derivatives
If we view x3 as the product of f(x)*g(x) where
f(x) = x2 and g(x) = x, then
(x3)' = (fg)' = (f)'g + f(g)' by the Product Rule
= (x2)'x + x2(x)' = (2x)x + x2(1)
so (x3)'= 3x2
If we view x4 as the product of f(x)*g(x) where
f(x) = x3 and g(x) = x and apply the result above,
then we get (x4)' = 4x3.
76. Computations of Derivatives
If we view x3 as the product of f(x)*g(x) where
f(x) = x2 and g(x) = x, then
(x3)' = (fg)' = (f)'g + f(g)' by the Product Rule
= (x2)'x + x2(x)' = (2x)x + x2(1)
so (x3)'= 3x2
If we view x4 as the product of f(x)*g(x) where
f(x) = x3 and g(x) = x and apply the result above,
then we get (x4)' = 4x3.
Continue in this manner (x5)' = 5x4, (x6)' = 6x5 etc..
77. Computations of Derivatives
If we view x3 as the product of f(x)*g(x) where
f(x) = x2 and g(x) = x, then
(x3)' = (fg)' = (f)'g + f(g)' by the Product Rule
= (x2)'x + x2(x)' = (2x)x + x2(1)
so (x3)'= 3x2
If we view x4 as the product of f(x)*g(x) where
f(x) = x3 and g(x) = x and apply the result above,
then we get (x4)' = 4x3.
Continue in this manner (x5)' = 5x4, (x6)' = 6x5 etc..
we obtain the formula of derivatives of the monomials.
Derivatives of Monomials
(xN)' =
78. Computations of Derivatives
If we view x3 as the product of f(x)*g(x) where
f(x) = x2 and g(x) = x, then
(x3)' = (fg)' = (f)'g + f(g)' by the Product Rule
= (x2)'x + x2(x)' = (2x)x + x2(1)
so (x3)'= 3x2
If we view x4 as the product of f(x)*g(x) where
f(x) = x3 and g(x) = x and apply the result above,
then we get (x4)' = 4x3.
Continue in this manner (x5)' = 5x4, (x6)' = 6x5 etc..
we obtain the formula of derivatives of the monomials.
Derivatives of Monomials
(xN)' = NxN–1, N = 0, 1, 2, ..
79. Computations of Derivatives
If we view x3 as the product of f(x)*g(x) where
f(x) = x2 and g(x) = x, then
(x3)' = (fg)' = (f)'g + f(g)' by the Product Rule
= (x2)'x + x2(x)' = (2x)x + x2(1)
so (x3)'= 3x2
If we view x4 as the product of f(x)*g(x) where
f(x) = x3 and g(x) = x and apply the result above,
then we get (x4)' = 4x3.
Continue in this manner (x5)' = 5x4, (x6)' = 6x5 etc..
we obtain the formula of derivatives of the monomials.
Derivatives of Monomials
(xN)' = NxN–1, N = 0, 1, 2, .. and that (cxN)' = cNxN–1
where c is a constant.
81. Computations of Derivatives
In the
dxN
dx
notation,
d
dx
= NxN–1
Example A. Find the derivatives of the following
functions.
a. –2x7– πx3 + sin(π/4)x2 + ln(8)
82. Computations of Derivatives
In the
dxN
dx
notation,
d
dx
= NxN–1
Example A. Find the derivatives of the following
functions.
a. –2x7– πx3 + sin(π/4)x2 + ln(8)
(–2x7– πx3 + sin(π/4)x2 + ln(8))'
83. Computations of Derivatives
In the
dxN
dx
notation,
d
dx
= NxN–1
Example A. Find the derivatives of the following
functions.
a. –2x7– πx3 + sin(π/4)x2 + ln(8)
(–2x7– πx3 + sin(π/4)x2 + ln(8))'
=(–2x7)' – (πx3)' + (sin(π/4)x2)' + (ln(8))'
84. Computations of Derivatives
In the
dxN
dx
notation,
d
dx
= NxN–1
Example A. Find the derivatives of the following
functions.
a. –2x7– πx3 + sin(π/4)x2 + ln(8)
(–2x7– πx3 + sin(π/4)x2 + ln(8))'
=(–2x7)' – (πx3)' + (sin(π/4)x2)' + (ln(8))'
= –2 (x7)' –π (x3)' + sin(π/4) (x2)' + 0
85. Computations of Derivatives
In the
dxN
dx
notation,
d
dx
= NxN–1
Example A. Find the derivatives of the following
functions.
a. –2x7– πx3 + sin(π/4)x2 + ln(8)
(–2x7– πx3 + sin(π/4)x2 + ln(8))'
=(–2x7)' – (πx3)' + (sin(π/4)x2)' + (ln(8))'
= –2 (x7)' –π (x3)' + sin(π/4) (x2)' + 0
= –2 (7x6) –π (3x2) + sin(π/4) 2x + 0
86. Computations of Derivatives
In the
dxN
dx
notation,
d
dx
= NxN–1
Example A. Find the derivatives of the following
functions.
a. –2x7– πx3 + sin(π/4)x2 + ln(8)
(–2x7– πx3 + sin(π/4)x2 + ln(8))'
=(–2x7)' – (πx3)' + (sin(π/4)x2)' + (ln(8))'
= –2 (x7)' –π (x3)' + sin(π/4) (x2)' + 0
= –2 (7x6) –π (3x2) + sin(π/4) 2x + 0
= –14x6 – 3πx2 + (√2/2)2x
87. Computations of Derivatives
In the
dxN
dx
notation,
d
dx
= NxN–1
Example A. Find the derivatives of the following
functions.
a. –2x7– πx3 + sin(π/4)x2 + ln(8)
(–2x7– πx3 + sin(π/4)x2 + ln(8))'
=(–2x7)' – (πx3)' + (sin(π/4)x2)' + (ln(8))'
= –2 (x7)' –π (x3)' + sin(π/4) (x2)' + 0
= –2 (7x6) –π (3x2) + sin(π/4) 2x + 0
= –14x6 – 3πx2 + (√2/2)2x
= –14x6 – 3πx2 + x√2
104. More Computations of Derivatives
The formulas for the derivatives of monomials may
be extended to the power function f(x) = xP where P is
any nonzero real number.
105. More Computations of Derivatives
The formulas for the derivatives of monomials may
be extended to the power function f(x) = xP where P is
any nonzero real number.
Derivatives of the Power Functions
106. More Computations of Derivatives
The formulas for the derivatives of monomials may
be extended to the power function f(x) = xP where P is
any nonzero real number.
Derivatives of the Power Functions
(xP)' = PxP–1 where P is a non–zero number.
107. More Computations of Derivatives
The formulas for the derivatives of monomials may
be extended to the power function f(x) = xP where P is
any nonzero real number.
Derivatives of the Power Functions
(xP)' = PxP–1 where P is a non–zero number.
Example B.
5
Find the derivative of f(x) = √4x2
108. More Computations of Derivatives
The formulas for the derivatives of monomials may
be extended to the power function f(x) = xP where P is
any nonzero real number.
Derivatives of the Power Functions
(xP)' = PxP–1 where P is a non–zero number.
Example B.
5
Find the derivative of f(x) = √4x2
5
(√4x2)'
= [(4x2)1/5] '
109. More Computations of Derivatives
The formulas for the derivatives of monomials may
be extended to the power function f(x) = xP where P is
any nonzero real number.
Derivatives of the Power Functions
(xP)' = PxP–1 where P is a non–zero number.
Example B.
5
Find the derivative of f(x) = √4x2
5
(√4x2)'
= [(4x2)1/5] '
= [41/5x2/5] '
110. More Computations of Derivatives
The formulas for the derivatives of monomials may
be extended to the power function f(x) = xP where P is
any nonzero real number.
Derivatives of the Power Functions
(xP)' = PxP–1 where P is a non–zero number.
Example B.
5
Find the derivative of f(x) = √4x2
5
(√4x2)'
= [(4x2)1/5] '
= [41/5x2/5] '
= 41/5 [x2/5] '
111. More Computations of Derivatives
The formulas for the derivatives of monomials may
be extended to the power function f(x) = xP where P is
any nonzero real number.
Derivatives of the Power Functions
(xP)' = PxP–1 where P is a non–zero number.
Example B.
5
Find the derivative of f(x) = √4x2
5
(√4x2)'
= [(4x2)1/5] '
= [41/5x2/5] '
= 41/5 [x2/5] '
= 41/5 [ 2 x2/5 – 1]
5
112. More Computations of Derivatives
The formulas for the derivatives of monomials may
be extended to the power function f(x) = xP where P is
any nonzero real number.
Derivatives of the Power Functions
(xP)' = PxP–1 where P is a non–zero number.
Example B.
5
Find the derivative of f(x) = √4x2
5
(√4x2)'
= [(4x2)1/5] '
= [41/5x2/5] '
= 41/5 [x2/5] '
= 41/5 [ 2 x2/5 – 1]
5
(41/5)
= 2 x –3/5
5