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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![**Problem Statement:**
Let \( f(x) = \left( \ln(x) \right)^{\sec(x)} \). Find \( f'(x) \).
**Solution:**
\[
f'(x) = \sec(x) \tan(x) \ln(\ln(x)) + \frac{\sec(x)}{x \ln(x)} (\ln(x))^{\sec(x)}
\]
**Details:**
- The image contains an equation differentiating a function where \( f(x) \) is given as \(\left( \ln(x) \right)^{\sec(x)}\).
- The calculation involves using the chain rule and product rule to find the derivative \( f'(x) \).
- The expression is initially entered in a text format and is then previewed in a more readable mathematical typeface.
**Explanation:**
- The derivative \( f'(x) \) includes the use of secant and tangent functions along with natural logarithms.
- The first term \( \sec(x) \tan(x) \ln(\ln(x)) \) involves the derivative of the outer function.
- The second term \(\frac{\sec(x)}{x \ln(x)} (\ln(x))^{\sec(x)}\) represents the derivative of the inner function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc84364e-ef46-429b-9d16-57e28333362b%2F70a76305-5e8d-495a-9274-1544b4db56e7%2F1v2bgn_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( f(x) = \left( \ln(x) \right)^{\sec(x)} \). Find \( f'(x) \).
**Solution:**
\[
f'(x) = \sec(x) \tan(x) \ln(\ln(x)) + \frac{\sec(x)}{x \ln(x)} (\ln(x))^{\sec(x)}
\]
**Details:**
- The image contains an equation differentiating a function where \( f(x) \) is given as \(\left( \ln(x) \right)^{\sec(x)}\).
- The calculation involves using the chain rule and product rule to find the derivative \( f'(x) \).
- The expression is initially entered in a text format and is then previewed in a more readable mathematical typeface.
**Explanation:**
- The derivative \( f'(x) \) includes the use of secant and tangent functions along with natural logarithms.
- The first term \( \sec(x) \tan(x) \ln(\ln(x)) \) involves the derivative of the outer function.
- The second term \(\frac{\sec(x)}{x \ln(x)} (\ln(x))^{\sec(x)}\) represents the derivative of the inner function.
Expert Solution
Basic differentiation properties
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