Chapter 11: Chapter 11

Problem 10

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Calculate the derivatives of the functions in Exercises 1-46. HINT [See Example 1.] \(f(x)=(x+7)^{-2}\)

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The derivative of the function \(f(x) = (x+7)^{-2}\) is \(f'(x) = -2(x+7)^{-3}\).

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TABLE OF CONTENTS

Step 1: Rewrite the function

Rewrite the function as: \(f(x) = (x+7)^{-2}\).

Step 2: Identify the outer and inner functions

In this case, the outer function is \(g(x) = x^{-2}\), and the inner function is \(h(x) = x+7\). Our goal is to find the derivative \(f'(x) = \frac{d}{dx} ((x+7)^{-2})\) by applying the chain rule.

Step 3: Apply the Chain Rule for Derivatives

Using the chain rule, which states \(\frac{d}{dx} (g(h(x))) = g'(h(x))h'(x)\), we will find the derivative of the outer function and the inner function: Outer function derivative: \(g'(x) = \frac{d}{dx} (x^{-2}) = -2x^{-3}\). Inner function derivative: \(h'(x) = \frac{d}{dx}(x+7) = 1\).

Step 4: Substitute and Simplify

Substitute the derivatives we found in step 3 into the chain rule formula: \(f'(x) = g'(h(x))h'(x) = (-2(x+7)^{-3})(1) = -2(x+7)^{-3}\). The derivative of the given function is: \(f'(x) = -2(x+7)^{-3}\).

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Chapter 11: Chapter 11

Calculate the derivatives of the functions in Exercises 1-46. HINT [See Example 1.] \(f(x)=(x+7)^{-2}\)

Short Answer

Step by step solution

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Step 1: Rewrite the function

Step 2: Identify the outer and inner functions

Step 3: Apply the Chain Rule for Derivatives

Step 4: Substitute and Simplify

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More chapters from the book ‘Finite Mathematics and Applied Calculus’

Chapter 0

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

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