Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 17

Chapter 3, Problem 17

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.
y = sin⁵x

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Step 1: Identify the composite function structure. The given function is y = \(\sin\)^5(x), which can be rewritten as y = (\(\sin\)(x))^5.

Step 2: Determine the inner function u = g(x). In this case, the inner function is u = \(\sin\)(x).

Step 3: Determine the outer function y = f(u). The outer function is y = u^5.

Step 4: Differentiate the outer function with respect to u. The derivative of y = u^5 with respect to u is \(\frac{dy}{du}\) = 5u^4.

Step 5: Differentiate the inner function with respect to x. The derivative of u = \(\sin\)(x) with respect to x is \(\frac{du}{dx}\) = \(\cos\)(x).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Composite Functions

A composite function is formed when one function is applied to the result of another function. In the context of the question, we need to identify an inner function g(x) and an outer function f(u) such that the overall function can be expressed as y = f(g(x)). Understanding how to break down a function into its components is essential for differentiation.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if y = f(g(x)), then the derivative dy/dx can be calculated as dy/dx = f'(g(x)) * g'(x). This rule allows us to find the derivative of complex functions by differentiating the outer function and multiplying it by the derivative of the inner function.

Power Rule

The power rule is a basic differentiation rule that states if y = x^n, then dy/dx = n*x^(n-1). In the given function y = sin⁵(x), recognizing that this is a power function of sin(x) is crucial. Applying the power rule in conjunction with the chain rule will help in finding the derivative of the composite function effectively.

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Power Rules

Related Practice

Textbook Question

Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.

Determine the acceleration of the object when its velocity is zero.

f(t) = 2t² - 9t + 12; 0 ≤ t ≤ 3

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Textbook Question

On what intervals is the speed increasing?

f(t) = t² - 4t; 0 ≤ t ≤ 5

Textbook Question

Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.

f(x) = 1/x; P(-1,-1)

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Textbook Question

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.

y = (5x²+11x)^4/3

Textbook Question

On what intervals is the speed increasing?

f(t) = 2t² - 9t + 12; 0 ≤ t ≤ 3

Textbook Question

Determine the acceleration of the object when its velocity is zero.

f(t) = t² - 4t; 0 ≤ t ≤ 5

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.y = sin⁵x

Verified video answer for a similar problem:

Key Concepts

Composite Functions

Chain Rule

Power Rule

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.
y = sin⁵x