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cos.x Use logarithmic differentiation to find y' for y=(Inx)*** . Answer needs to in terms of x only!

cos.x Use logarithmic differentiation to find y' for y=(Inx)*** . Answer needs to in terms of x only!

Calculus: Early Transcendentals
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:** Use logarithmic differentiation to find \( y' \) for \( y = (\ln x)^{\cos x} \). Answer needs to be in terms of \( x \) only! **Solution Overview:** To solve this problem, we need to utilize logarithmic differentiation since the function is a combination of logarithmic and trigonometric functions raised to a power. By taking the natural log on both sides, we can simplify the differentiation process. Here are the steps we'll follow: 1. Take the natural logarithm of both sides: \(\ln y = \ln((\ln x)^{\cos x})\). 2. Use the logarithm power rule: \(\ln y = \cos x \cdot \ln(\ln x)\). 3. Differentiate both sides with respect to \(x\): - The left side becomes \(\frac{1}{y} \cdot y'\). - The right side requires the product rule and chain rule. 4. Solve for \(y'\) and express it in terms of \(x\) only. The result will give you the derivative of the function \( y = (\ln x)^{\cos x} \) in terms of \( x \).
Transcribed Image Text:**Problem Statement:** Use logarithmic differentiation to find \( y' \) for \( y = (\ln x)^{\cos x} \). Answer needs to be in terms of \( x \) only! **Solution Overview:** To solve this problem, we need to utilize logarithmic differentiation since the function is a combination of logarithmic and trigonometric functions raised to a power. By taking the natural log on both sides, we can simplify the differentiation process. Here are the steps we'll follow: 1. Take the natural logarithm of both sides: \(\ln y = \ln((\ln x)^{\cos x})\). 2. Use the logarithm power rule: \(\ln y = \cos x \cdot \ln(\ln x)\). 3. Differentiate both sides with respect to \(x\): - The left side becomes \(\frac{1}{y} \cdot y'\). - The right side requires the product rule and chain rule. 4. Solve for \(y'\) and express it in terms of \(x\) only. The result will give you the derivative of the function \( y = (\ln x)^{\cos x} \) in terms of \( x \).
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