The Chain Rule Powerpoint Lesson
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The document explains the chain rule, which provides a method for finding the derivative of a composite function. The chain rule states that the derivative of a composite function f(g(x)) is equal to the derivative of the outside function f'(g(x)) multiplied by the derivative of the inside function g'(x). This allows the calculation of derivatives of more complex functions that cannot be solved using basic derivative rules. Several examples are provided to demonstrate how to use the chain rule to calculate derivatives of various composite functions.
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Lesson 11: The Chain Rule
The chain rule allows us to differentiate a composition of functions. It's a complicated peel-the-onion process, but it can be learned.
Chain rule
1. The chain rule describes how to take the derivative of a function composed of other functions. It states that if z = f(x, y) and x and y are functions of t, then the derivative of z with respect to t is the sum of the partial derivatives multiplied by the derivatives of x and y with respect to t.
2. The chain rule is applied to find the derivative of two example functions, w = xy and w = xy + z, with respect to t along given paths for x, y, and z in terms of t.
3. The chain rule is generalized to the case of a function u of n variables, where each variable is a function of m other variables
The chain rule
The document discusses the chain rule and how to use it to differentiate and integrate composite functions. The chain rule states that if h(x) = g(f(x)), then h'(x) = g'(f(x))f'(x). It provides examples of applying the chain rule to differentiate functions like sin(x2 - 4) and integrate functions like ∫(3x2 + 4)3 dx. It also discusses how to integrate functions of the form f'(x)g(f(x)) by recognizing them as derivatives of composite functions.
Chain Rule
The document discusses the chain rule and how to use it to find derivatives of more complex equations. It provides examples of using the chain rule to take derivatives of functions involving exponents, trigonometric functions, radicals, and combinations of these. Key steps include identifying the inner and outer functions, taking the derivative of the inner function, and plugging into the chain rule formula. The document also contrasts using the chain rule method versus the inside-outside method for some problems.
Chain Rule
The document discusses the chain rule, which is used to find the derivative of composite functions. It provides examples of applying the chain rule to functions of the form f(g(x)) by taking the derivative of the outside function with respect to the inside function, and multiplying by the derivative of the inside function with respect to x. The chain rule can be used repeatedly when a function is composed of multiple nested functions. Derivative formulas themselves incorporate the chain rule. The chain rule is essential for finding derivatives and is the most common mistake made by students on tests.
Lesson 10: The Chain Rule
The chain rule helps us find a derivative of a composition of functions. It turns out that it's the product of the derivatives of the composed functions.
The chain rule
1) The document explains the chain rule for differentiating composite functions.
2) It provides examples of applying the chain rule to functions of the form f(g(x)) where f and g are different functions.
3) The key steps are to let the inner function g(x) equal a variable u, take the derivative of the outer function f(u) with respect to u, and multiply it by the derivative of the inner function g(x) with respect to x.
Engineering Mathematics - Total derivatives, chain rule and derivative of imp...
This document discusses the total derivative and methods for finding derivatives of functions with multiple variables.
The total derivative expresses the total differential of a function u with respect to time t as the sum of the partial derivatives of u with respect to each variable x1, x2,...xn, multiplied by the rate of change of that variable with respect to time.
The chain rule is used to take derivatives of composite functions, where the output of one function is an input to another. The derivative is expressed as the product of the partial derivatives of each nested function.
Derivatives can also be taken for implicit functions, where not all variables can be solved for explicitly. The derivative of one variable with respect to another in an
Recommended
Lesson 11: The Chain Rule
The chain rule allows us to differentiate a composition of functions. It's a complicated peel-the-onion process, but it can be learned.
Chain rule
1. The chain rule describes how to take the derivative of a function composed of other functions. It states that if z = f(x, y) and x and y are functions of t, then the derivative of z with respect to t is the sum of the partial derivatives multiplied by the derivatives of x and y with respect to t.
2. The chain rule is applied to find the derivative of two example functions, w = xy and w = xy + z, with respect to t along given paths for x, y, and z in terms of t.
3. The chain rule is generalized to the case of a function u of n variables, where each variable is a function of m other variables
The chain rule
The document discusses the chain rule and how to use it to differentiate and integrate composite functions. The chain rule states that if h(x) = g(f(x)), then h'(x) = g'(f(x))f'(x). It provides examples of applying the chain rule to differentiate functions like sin(x2 - 4) and integrate functions like ∫(3x2 + 4)3 dx. It also discusses how to integrate functions of the form f'(x)g(f(x)) by recognizing them as derivatives of composite functions.
Chain Rule
The document discusses the chain rule and how to use it to find derivatives of more complex equations. It provides examples of using the chain rule to take derivatives of functions involving exponents, trigonometric functions, radicals, and combinations of these. Key steps include identifying the inner and outer functions, taking the derivative of the inner function, and plugging into the chain rule formula. The document also contrasts using the chain rule method versus the inside-outside method for some problems.
Chain Rule
The document discusses the chain rule, which is used to find the derivative of composite functions. It provides examples of applying the chain rule to functions of the form f(g(x)) by taking the derivative of the outside function with respect to the inside function, and multiplying by the derivative of the inside function with respect to x. The chain rule can be used repeatedly when a function is composed of multiple nested functions. Derivative formulas themselves incorporate the chain rule. The chain rule is essential for finding derivatives and is the most common mistake made by students on tests.
Lesson 10: The Chain Rule
The chain rule helps us find a derivative of a composition of functions. It turns out that it's the product of the derivatives of the composed functions.
The chain rule
1) The document explains the chain rule for differentiating composite functions.
2) It provides examples of applying the chain rule to functions of the form f(g(x)) where f and g are different functions.
3) The key steps are to let the inner function g(x) equal a variable u, take the derivative of the outer function f(u) with respect to u, and multiply it by the derivative of the inner function g(x) with respect to x.
Engineering Mathematics - Total derivatives, chain rule and derivative of imp...
This document discusses the total derivative and methods for finding derivatives of functions with multiple variables.
The total derivative expresses the total differential of a function u with respect to time t as the sum of the partial derivatives of u with respect to each variable x1, x2,...xn, multiplied by the rate of change of that variable with respect to time.
The chain rule is used to take derivatives of composite functions, where the output of one function is an input to another. The derivative is expressed as the product of the partial derivatives of each nested function.
Derivatives can also be taken for implicit functions, where not all variables can be solved for explicitly. The derivative of one variable with respect to another in an
4.1 the chain rule
The document discusses the chain rule for taking derivatives of composite functions. It explains that the chain rule allows you to break down composite functions into their constituent parts. The chain rule states that the derivative of the outside function is the derivative of the inside function multiplied by the derivative of the inside with respect to the outside. The document provides an example of using the chain rule to take the derivative of a polynomial function composed of simpler functions. It also notes that the general power rule can be used to take derivatives when the outside function is a power and the inside is a differentiable function.
Derivatives vinnie
The document outlines various derivative rules including:
- The power rule for derivatives of functions with exponents like x^n.
- Sum and difference rules for derivatives of sums and differences of functions.
- Product and quotient rules for derivatives of products and quotients of functions.
- The chain rule for derivatives of composite functions like f(g(x)).
- Derivative rules for common trigonometric, exponential and logarithmic functions.
Examples are provided to demonstrate how to apply each rule to find derivatives.
The Chain Rule Powerpoint Lesson
The document explains the chain rule, which provides a method for finding the derivative of a composite function. The chain rule states that the derivative of a composite function f(g(x)) is equal to the derivative of the outer function f multiplied by the derivative of the inner function g. This can be written as f'(g(x)) * g'(x). Several examples are provided to demonstrate how to use the chain rule to take the derivative of various composite functions.
The Chain Rule Powerpoint Lesson
The document explains the chain rule, which provides a method for finding the derivative of a composite function. The chain rule states that the derivative of a composite function f(g(x)) is equal to the derivative of the outer function f multiplied by the derivative of the inner function g. This can be written as f'(g(x)) * g'(x). Several examples are provided to demonstrate how to use the chain rule to take the derivative of various composite functions.
Implicit differentiation
To summarize:
1) To find the derivative of an implicit function y=y(x) defined by an equation F(x,y)=0, take the derivative of both sides with respect to x.
2) This will give a new equation involving x, y, and dy/dx that can be solved for dy/dx.
3) The examples show applying this process to find derivatives and tangent lines for various implicit equations.
Understanding the remainder theorem
The document discusses using the remainder theorem to find the remainder of a polynomial function divided by a binomial function without fully performing the division. It provides two examples:
1) When f(x) = x^3 - 3x^2 + 7 is divided by g(x) = x + 2, the remainder is -13.
2) When f(x) = x^3 - 3x^2 + 5x - 1 is divided by g(x) = x - 1, the remainder is 2.
The key steps are to set the binomial factor equal to 0 to find the value to substitute into the polynomial, then evaluate the polynomial at that value to determine the remainder.
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Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
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3) Additional examples include differentiating trigonometric, exponential, and other functions with respect to x.
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This document discusses solving for the area under the curve of a function algebraically without using a graphing calculator. It first finds the derivative of the function f(x) = ln(2x^2 + 3) using the chain rule. It then takes the integral of the derivative from 0 to 4 to find the area beneath the curve, which equals 2.4567.
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The document discusses the chain rule for taking derivatives of composite functions. It explains that the chain rule allows you to break down composite functions into their constituent parts. The chain rule states that the derivative of the outside function is the derivative of the inside function multiplied by the derivative of the inside with respect to the outside. The document provides an example of using the chain rule to take the derivative of a polynomial function composed of simpler functions. It also notes that the general power rule can be used to take derivatives when the outside function is a power and the inside is a differentiable function.
Derivatives vinnie
The document outlines various derivative rules including:
- The power rule for derivatives of functions with exponents like x^n.
- Sum and difference rules for derivatives of sums and differences of functions.
- Product and quotient rules for derivatives of products and quotients of functions.
- The chain rule for derivatives of composite functions like f(g(x)).
- Derivative rules for common trigonometric, exponential and logarithmic functions.
Examples are provided to demonstrate how to apply each rule to find derivatives.
The Chain Rule Powerpoint Lesson
The document explains the chain rule, which provides a method for finding the derivative of a composite function. The chain rule states that the derivative of a composite function f(g(x)) is equal to the derivative of the outer function f multiplied by the derivative of the inner function g. This can be written as f'(g(x)) * g'(x). Several examples are provided to demonstrate how to use the chain rule to take the derivative of various composite functions.
The Chain Rule Powerpoint Lesson
The document explains the chain rule, which provides a method for finding the derivative of a composite function. The chain rule states that the derivative of a composite function f(g(x)) is equal to the derivative of the outer function f multiplied by the derivative of the inner function g. This can be written as f'(g(x)) * g'(x). Several examples are provided to demonstrate how to use the chain rule to take the derivative of various composite functions.
Implicit differentiation
To summarize:
1) To find the derivative of an implicit function y=y(x) defined by an equation F(x,y)=0, take the derivative of both sides with respect to x.
2) This will give a new equation involving x, y, and dy/dx that can be solved for dy/dx.
3) The examples show applying this process to find derivatives and tangent lines for various implicit equations.
Understanding the remainder theorem
The document discusses using the remainder theorem to find the remainder of a polynomial function divided by a binomial function without fully performing the division. It provides two examples:
1) When f(x) = x^3 - 3x^2 + 7 is divided by g(x) = x + 2, the remainder is -13.
2) When f(x) = x^3 - 3x^2 + 5x - 1 is divided by g(x) = x - 1, the remainder is 2.
The key steps are to set the binomial factor equal to 0 to find the value to substitute into the polynomial, then evaluate the polynomial at that value to determine the remainder.
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Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
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This document explains the chain rule for differentiating composite functions. The chain rule states that if y = u(x), where u is expressed in terms of x, then dy/dx = (dy/du) * (du/dx). The document provides examples of applying the chain rule to differentiate functions like y = (3x + 2)^4 and y = 1/(x^2 + 1).
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1) It provides the formula for differentiating y=u/v with respect to x and works through an example.
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3) Additional examples include differentiating trigonometric, exponential, and other functions with respect to x.
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0210 ch 2 day 10
The document discusses combining functions through addition, subtraction, multiplication, and division. Some key points:
- Functions can be combined using the same rules as algebraic expressions, such as f(x) + g(x) = (f + g)(x).
- An example demonstrates combining two functions f(x) and g(x) through addition, subtraction, multiplication, and division.
- Composite functions are discussed where the output of one function acts as the input for another function, written as f(g(x)).
- Several examples demonstrate evaluating composite functions by first evaluating the inner function and then the outer function.
- It is shown that f(g(x)) is not
Operation on functions
The document discusses operations that can be performed on functions, including addition, subtraction, multiplication, division, and composition. It provides examples of evaluating each type of operation on functions by first evaluating the individual functions at a given value or variable and then performing the indicated operation on the results. Composition involves evaluating the inner function first and substituting its result into the outer function.
Algebra 2 Section 5-1
The document discusses operations that can be performed on functions, including addition, subtraction, multiplication, and division. Definitions of each operation are provided, along with examples of applying the operations to specific functions. Addition of functions involves adding the outputs of each function, subtraction involves subtracting the outputs, multiplication involves multiplying the outputs, and division involves dividing the outputs given the denominator function is not equal to 0. Several examples are worked through applying the different operations to functions like f(x)=2x and g(x)=-x+5. The examples also demonstrate evaluating composite functions and restricting domains as needed.
01. Differentiation-Theory & solved example Module-3.pdf
Total No. of questions in Differentiation are-
In Chapter Examples 31
Solved Examples 32
The rate of change of one quantity with respect to some another quantity has a great importance. For example the rate of change of displacement of a particle with respect to time is called its velocity and the rate of change of velocity is
called its acceleration.
The following results can easily be established using the above definition of the derivative–
d
(i) dx (constant) = 0
The rate of change of a quantity 'y' with respect to another quantity 'x' is called the derivative or differential coefficient of y with respect to x.
Let y = f(x) be a continuous function of a variable quantity x, where x is independent and y is
(ii)
(iii)
(iv)
(v)
d
dx (ax) = a
d (xn) = nxn–1
dx
d ex =ex
dx
d (ax) = ax log a
dependent variable quantity. Let x be an arbitrary small change in the value of x and y be the
dx
d
(vi) dx
e
(logex) = 1/x
corresponding change in y then lim
y
if it exists, d 1
x0 x
is called the derivative or differential coefficient of y with respect to x and it is denoted by
(vii) dx
(logax) =
x log a
dy . y', y
dx 1
or Dy.
d
(viii) dx (sin x) = cos x
So, dy dx
dy
dx
lim
x0
lim
x0
y
x
f (x x) f (x)
x
(ix) (ix)
(x) (x)
d
dx (cos x) = – sin x
d (tan x) = sec2x
dx
The process of finding derivative of a function is called differentiation.
If we again differentiate (dy/dx) with respect to x
(xi)
d (cot x) = – cosec2x
dx
d
then the new derivative so obtained is called second derivative of y with respect to x and it is
Fd2 y
(xii) dx
d
(xiii) dx
(secx)= secx tan x
(cosec x) = – cosec x cot x
denoted by
HGdx2 Jor y" or y2 or D2y. Similarly,
d 1
we can find successive derivatives of y which
(xiv) dx
(sin–1 x) = , –1< x < 1
1 x2
may be denoted by
d –1 1
d3 y d4 y
dn y
(xv) dx (cos x) = –
,–1 < x < 1
dx3 ,
dx4 , ........, dxn , ......
d
(xvi) dx
(tan–1 x) = 1
1 x2
Note : (i)
y is a ratio of two quantities y and
x
(xvii) (xvii)
d (cot–1 x) = – 1
where as dy
dx
dy
is not a ratio, it is a single
dx
d
(xviii) (xviii)
(sec–1 x) =
1 x2
1
|x| > 1
quantity i.e.
dx dy÷ dx
dx x x2 1
(ii)
dy is
dx
d (y) in which d/dx is simply a symbol
dx
(xix)
d (cosec–1 x) = – 1
dx
of operation and not 'd' divided by dx.
d
(xx) dx
(sinh x) = cosh x
d
(xxi) dx
d
(cosh x) = sinh x
Theorem V Derivative of the function of the function. If 'y' is a function of 't' and t' is a function of 'x' then
(xxii) dx
d
(tanh x) = sech2 x
dy =
dx
dy . dt
dt dx
(xxiii) dx
d
(xxiv) dx
d
(coth x) = – cosec h2 x (sech x) = – sech x tanh x
Theorem VI Derivative of parametric equations If x = (t) , y = (t) then
dy dy / dt
=
(xxv) dx
(cosech x) = – cosec hx coth x
dx dx / dt
(xxvi) (xxvi)
(xxvii) (xxvii)
d (sin h–1 x) =
maths basics
This document provides formulas for taking derivatives and integrals of common functions. There are 22 differentiation formulas that give the derivatives of functions like constants, sums and differences of functions, composite functions, polynomials, trigonometric functions and their inverses, exponential functions, and logarithmic functions. There are also 19 integration formulas that give the integrals of polynomials, rational functions, trigonometric functions and their inverses, exponential functions, and logarithmic functions.
Dif int
This document provides formulas for taking derivatives and integrals of common functions. There are 22 differentiation formulas that give the derivatives of functions like constants, sums and differences of functions, composite functions, polynomials, trigonometric functions and their inverses, exponential functions, and logarithmic functions. There are also 19 integration formulas that give the integrals of polynomials, rational functions, trigonometric functions and their inverses, exponential functions, and logarithmic functions.
Comp inverse
This document discusses functions, function notation, function composition, and function inverses. It provides examples of:
1) Finding the value of a function by substituting inputs.
2) Composing functions by substituting one function into another.
3) Finding the inverse of a function by swapping inputs and outputs and solving for the new output.
Operations on Functions.pdf
This document discusses functions and operations that can be performed on functions. It defines a function as a relation that assigns each input exactly one output. The four basic operations that can be performed on functions are addition, subtraction, multiplication, and division. Examples are provided to demonstrate how to apply these operations by combining two functions using the notation for each operation. The key is to apply the operation to the outputs of each function using the same input. Several practice problems are then given for the student to work through.
functions limits and continuity
This document provides an overview of functions, limits, and continuity. It defines key concepts such as domain and range of functions, and examples of standard real functions. It also covers even and odd functions, and how to calculate limits, including left and right hand limits. Methods for evaluating algebraic limits using substitution, factorization, and rationalization are presented. The objectives are to understand functions, domains, ranges, and how to evaluate limits of functions.
FUNCIONES.pptx
This document contains exercises on operations with functions. It gives two functions, f(x) = x^2 + 2x - 3 and g(x) = x^2 + 3x - 4, and asks to find:
a) f+g(x)
b) f-g(x)
c) f*g(x)
d) f/g(x)
e) f(x+2)
f) f(1/4)
It also asks to find the composite function f o g(x) and its domain, where f(x) = x - 3 and g(x) = x^2 + 2. Finally,
Lesson 4 Nov 17 09
The document provides information on derivatives including:
- Rules for finding derivatives of sums, products, and quotients using the product, quotient, and chain rules
- Derivatives of common trigonometric functions like sine, cosine, tangent, cotangent, secant, and cosecant
- An exercise involving 23 problems calculating derivatives
125 7.7
This document discusses the chain rule for finding derivatives. It explains that the chain rule is needed when taking the derivative of a composition of functions, where an "inside function" is plugged into an "outside function". The chain rule formula is given as the derivative of the outside function multiplied by the derivative of the inside function. Several examples are worked through, applying the chain rule when the power rule alone cannot be used, such as when the base of an exponent is a function rather than a variable. The document also notes that problems may require using multiple derivative rules, like the product rule and chain rule, to fully solve them.
Product rule
- The product of two differentiable functions is itself differentiable. The derivative of the product of two functions f and g is equal to the derivative of the first function times the second function, plus the derivative of the second function times the first function.
- Gottfried Leibniz originally wrote the formula for the product rule, motivated by the expression (x+dx) (y+dy) - xy. Subtracting dx dy from this expression resulted in the traditional differential form of the product rule.
Functions limits and continuity
This document discusses functions, limits, and continuity. It begins by defining functions, domains, ranges, and some standard real functions like constant, identity, modulus, and greatest integer functions. It then covers limits of functions including one-sided limits and properties of limits. Examples are provided to illustrate evaluating limits using substitution and factorization methods. The overall objectives are to understand functions, domains, ranges, limits of functions and methods to evaluate limits.
7.3 power functions and function operations
A power function has the form y=ax^b where a is a real number and b is a rational number. Functions can be combined using addition, subtraction, multiplication, division, and composition. When combining functions, the domain is the set of values where both functions are defined. Composition involves applying one function to the output of another.
Basic Calculus.docx
The document provides an overview of basic calculus concepts including functions, trigonometric functions, operations on functions, and composite functions. It defines a function as a rule that assigns exactly one output value to each input value. It also defines domain as the set of permissible input values and range as the set of corresponding output values. Examples are provided to demonstrate evaluating functions, finding limits, graphing functions, and performing operations like addition, multiplication, and composition on functions.
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The Chain Rule Powerpoint Lesson
- 2. The Problem Complex Functions Why? not all derivatives can be found through the use of the power, product, and quotient rules
- 3. Working To A Solution Composite function f(g(x)) f is the outside function, g is the inside function z=g(x), y=f(z), and y=f(g(x)) Therefore, a small change in x leads to a small change in z a small change in z leads to a small change in y
- 4. Working To A Solution cont. Therefore, (∆y/∆x) = (∆y/∆z) (∆z/∆x) Since (dy/dx)=limx0 (∆y/∆x) (dy/dx) = (dy/dz) (dz/dx) This is known as “The Chain Rule”
- 6. Example:f(x) = [ (x^3) + 2x + 1 ]^3 inside function = z = g(x) = (x^3) + 2x + 1 outside function = f(z) = z^3 g’(x) = (3x^2) + 2 f’(z) = 3[z]^2 (d/dx) f(g(x)) = 3 [ (3x^2) + 2 ] [ z]^2 = [ (9x^2) + 6 ] [ (x^3) + 2x + 1 ]^2
- 7. Examplef(x) = e^(4x^2) inside function = z = g(x) = 4x^2 outside function = f(z) = e^z g’(x) = 8x f’(z) = e^z (d/dx) f(g(x)) = 8xe^z = 8xe^(4x^2)
- 8. Examplef(x) = [(x^3) + 1]^(1/2) Inside function = z = g(x) = (x^3) + 1 outside function = f(z) = z^(1/2) g’(x) = 3x^2 f’(z) = (1/2)z^(-1/2) (d/dx) f(g(x)) = (3x^2) (1/2)z^(-1/2) = [(3/2)x^2] [(x^3) + 1]^(-1/2)
- 9. Homework Chapter 3.4Problems: 1,2,4,5,6,7,8,10,11,12,15,16,18,20,27 Remember to show all work. Turn in the assignment before the beginning of the next class period.
- 10. Sources Slideshow created using Microsoft PowerPoint Clipart, themes, and sound supplied by Microsoft PowerPoint Mathematics reference and notation from Hughes-Hallett, , Gleason, McCallum, and et al. Calculus: Single and Multivariable. 4th. 2005. 127-130. Print.