Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Problem Statement**
Differentiate \( f(x) = \tan^{-1}(5x) \). Use exact values, and don't forget the chain rule.
**Solution:**
\( f'(x) = \) [placeholder box for the answer]
**Explanation:**
To solve this, we use the chain rule for differentiation. The chain rule is a formula to compute the derivative of a composite function. If you have a function \( f(g(x)) \), then its derivative is \( f'(g(x)) \cdot g'(x) \).
In this case, the function is \( f(x) = \tan^{-1}(5x) \).
1. **Outer function:** \( \tan^{-1}(u) \) where \( u = 5x \). The derivative of \( \tan^{-1}(u) \) with respect to \( u \) is \( \frac{1}{1+u^2} \).
2. **Inner function:** \( u = 5x \). The derivative of \( 5x \) with respect to \( x \) is 5.
Applying the chain rule:
\[ f'(x) = \left( \frac{1}{1+(5x)^2} \right) \cdot 5 \]
This simplifies to:
\[ f'(x) = \frac{5}{1+25x^2} \]
This derivative expresses the rate of change of the function \( f(x) = \tan^{-1}(5x) \) with respect to \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faea988ab-7cea-47a3-b3b5-8670682902d0%2F9e310a06-89a7-4d6f-9ed9-9dc94cfb5a08%2Frby575_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Differentiate \( f(x) = \tan^{-1}(5x) \). Use exact values, and don't forget the chain rule.
**Solution:**
\( f'(x) = \) [placeholder box for the answer]
**Explanation:**
To solve this, we use the chain rule for differentiation. The chain rule is a formula to compute the derivative of a composite function. If you have a function \( f(g(x)) \), then its derivative is \( f'(g(x)) \cdot g'(x) \).
In this case, the function is \( f(x) = \tan^{-1}(5x) \).
1. **Outer function:** \( \tan^{-1}(u) \) where \( u = 5x \). The derivative of \( \tan^{-1}(u) \) with respect to \( u \) is \( \frac{1}{1+u^2} \).
2. **Inner function:** \( u = 5x \). The derivative of \( 5x \) with respect to \( x \) is 5.
Applying the chain rule:
\[ f'(x) = \left( \frac{1}{1+(5x)^2} \right) \cdot 5 \]
This simplifies to:
\[ f'(x) = \frac{5}{1+25x^2} \]
This derivative expresses the rate of change of the function \( f(x) = \tan^{-1}(5x) \) with respect to \( x \).
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