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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 18
Chapter 3, Problem 18

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 = sin x⁵

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Step 1: Identify the composite function structure. The given function is y = \(\sin\)(x^5). This can be seen as a composition of two functions: an inner function and an outer function.
Step 2: Define the inner function u = g(x). In this case, choose the inner function as u = x^5. This simplifies the original function to y = \(\sin\)(u).
Step 3: Define the outer function y = f(u). With the inner function u = x^5, the outer function becomes y = \(\sin\)(u).
Step 4: Differentiate the outer function with respect to u. The derivative of y = \(\sin\)(u) with respect to u is \(\frac{dy}{du}\) = \(\cos\)(u).
Step 5: Differentiate the inner function with respect to x. The derivative of u = x^5 with respect to x is \(\frac{du}{dx}\) = 5x^4.

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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 express the function y = sin(x⁵) as a composition of two functions: an inner function g(x) = x⁵ and an outer function f(u) = sin(u). Understanding how to identify and separate these functions is crucial for applying the chain rule in differentiation.
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Chain Rule

The chain rule is a fundamental theorem 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 breaking them down into simpler parts, making it essential for solving the given problem.
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Intro to the Chain Rule

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In this problem, we need to differentiate the composite function y = sin(x⁵) using the chain rule. Understanding how to compute derivatives and apply differentiation techniques is vital for obtaining the final result of dy/dx.
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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

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

f(x) = 4/x2; P(-1,4)

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

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.

On what intervals is the speed increasing?

f(t) = 18t - 3t2; 0 ≤ t ≤ 8

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.

On what intervals is the speed increasing?

f(t) = 2t2 - 9t + 12; 0 ≤ t ≤ 3

Textbook Question

Position, velocity, and acceleration 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.

b. Find and graph the velocity function. When is the object stationary, moving to the right, and moving to the left?

f(t) = 18t-3t²; 0 ≤ t ≤ 8