4.1 the chain rule
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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.
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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.
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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.
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The document discusses the chain rule, which is used to find the derivative of a composite function. It states that if y changes dy/du times as fast as u, and u changes du/dx times as fast as x, then y changes (dy/du)(du/dx) times as fast as x. It provides examples of applying the chain rule to find derivatives of composite functions, including functions with multiple compositions. It also discusses using the general power rule as a special case of the chain rule and simplifying derived functions using algebra. Finally, it outlines the chain rule versions of derivatives for trigonometric functions.
Chain rule
The chain rule allows you to take the derivative of a composite function f(g(x)). It states that the derivative of f(g(x)) is f'(g(x)) * g'(x). You apply the chain rule by taking the derivative of the outside function first, and then taking the derivative of the inside function. The chain rule can also be used with the product rule and quotient rule when differentiating composite functions.
Chain rule
The chain rule allows you to take the derivative of a composite function f(g(x)). It states that the derivative of f(g(x)) is f'(g(x)) * g'(x). You apply the chain rule by taking the derivative of the outside function first, then taking the derivative of the inside function. The chain rule can also be used with the product rule and quotient rule when differentiating composite functions.
Integration - Mathematics - UoZ
Presentation about integration - mathematics - University of Zakho ... College of Engineering - Petroleum Department
Sample2
This document contains a math worksheet with 20 practice problems on integration using the substitution rule. It begins with an explanation of the substitution rule and its relationship to the chain rule. Students are asked to use substitutions to evaluate definite and indefinite integrals involving trigonometric, exponential, logarithmic and algebraic functions. The difficulty of the problems increases throughout the worksheet.
exponen dan logaritma
This document discusses derivatives of various functions including:
- Exponential functions like ex and ax where the derivative of ex is ex and the derivative of ax is axln(a)
- Inverse functions where the derivative of the inverse is the reciprocal of the derivative of the original function
- Logarithmic functions like ln(x), loga(x) where the derivatives are 1/x and 1/(xln(a))
- Using logarithmic differentiation to find derivatives of functions like f(x)g(x)
It also provides practice problems finding derivatives of various functions and solving related equations.
Formulas
Integration by substitution allows difficult integrals to be evaluated by making a substitution of variables that simplifies the integrand. This technique involves changing both the variable of integration and the limits of integration. Key steps include identifying an appropriate substitution, determining the differential, and performing the integration with respect to the new variable before substituting back to the original. Extensive practice is important to master integration by substitution.
Integration intro
This document introduces integration as the reverse process of differentiation. It establishes the rule that if dy/dx = axn, then the integral is y = axn+1/(n+1) + c, where c is the constant of integration. Several examples are worked out applying this rule. The document also introduces notation for integration, including the integral sign ∫ and discusses some cases where the power is not a whole number. It emphasizes that the power in the integral must be positive.
19 3
The document discusses solving second order linear differential equations with constant coefficients. It introduces the process of finding the complementary function, which is the general solution to the homogeneous equation. The complementary function contains two arbitrary constants. The document provides examples of finding the complementary function when the auxiliary equation has real roots, equal roots, and complex roots. When the auxiliary equation has complex roots k1 and k2, the complementary function is written as eαx(A cos βx + B sin βx), where α and β are the real and imaginary parts of k1 and k2.
Lec5_Product & Quotient Rule.ppt
The document discusses differentiation rules for products and quotients of functions. It begins by introducing the product rule, which states that the derivative of a product of two functions f and g is equal to f times the derivative of g plus g times the derivative of f. Next, it derives the quotient rule through a similar process, concluding that the derivative of a quotient of two functions u and v is equal to the denominator v times the derivative of the numerator u minus the numerator u times the derivative of the denominator v, all over the square of the denominator v squared. Several examples are provided to demonstrate applying these rules to find derivatives.
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.
Chapter 9 differentiation
This document provides an explanation of differentiation and examples of calculating limits and derivatives using the first principle definition of the derivative. It begins by defining the limit of a function and providing examples of evaluating limits. It then introduces the concept of the derivative as the slope of the tangent line to a curve and explains how to calculate derivatives using small changes in x and y. The document provides examples of finding derivatives using this first principle definition. It also discusses rules for deriving composite functions and products of polynomials. Exercises are provided throughout for students to practice differentiation.
Chapter 9- Differentiation Add Maths Form 4 SPM
This document provides an explanation of differentiation and examples of calculating limits and derivatives using the first principle definition of the derivative. It begins by defining the limit of a function and providing examples of evaluating limits. It then introduces the concept of the derivative as the slope of the tangent line to a curve and explains how to calculate derivatives using small changes in x and y. The document provides examples of finding derivatives using this first principle definition. It also discusses rules for deriving composite functions and products of polynomials. Exercises are provided throughout for students to practice differentiation.
Rules of derivative
This document discusses differentiation and derivatives. It defines differentiation as finding the average rate of change of one variable with respect to another. It then discusses various methods of finding derivatives, including the direct method using derivative rules, as well as discussing specific rules like the power rule, product rule, quotient rule, chain rule, and rules for derivatives of trigonometric, exponential, and logarithmic functions.
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College textbook business math and statistics - section b - business mathem...
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4.1 the chain rule
- 1. Warm-up Find the derivative of the following: 1) 2) 3) 13 2 x 2 23 x x2 sin
- 3. The Chain Rule Derivatives become complicated when we have composite functions Use a substitution, u = “the inside function” then Break up functions using the chain rule: 253 2 xxu dx du du dy dx dy “derivative of y in terms of u” “derivative of u in terms of x” 42 253 xxy 4 uy
- 4. 56 253 2 x dx du xxu 3 4 4u du dy uy 𝑑𝑦 𝑑𝑥 = 𝑑𝑦 𝑑𝑢 ∙ 𝑑𝑢 𝑑𝑥 𝑑𝑦 𝑑𝑥 = 4𝑢3 ∙ 6𝑥 − 5 𝑑𝑦 𝑑𝑥 = 4 3𝑥2 − 5𝑥 + 2 3 ∙ 6𝑥 − 5
- 5. Steps: 1) Set u equal to an “inner” function 2) Rewrite original function in terms of u 3) Derive u in terms of x and y in terms of u 4) Plug derivatives into chain rule 5) Simplify Function form: xgxgfxgf dx d
- 6. 42 253 xxy dx dy
- 7. Example 1 Find the derivatives of the warm-ups using the chain rule 13 2 x
- 8. 2 23 x
- 9. x2 sin
- 11. Since polynomials are prevalent we come up with a general power rule: derivative of outside function times derivative of inside function dx dy
- 12. Example 1 Find the derivative
- 14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slopes of Parametrized Curves A parametrized curve , is if and are differentiable at . x t y t differentiable at t x y t
- 15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 3- 15 Finding dy/dx Parametrically If all three derivatives exist and 0, dx dt dy dy dt dxdx dt