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 the derivative of the function.
Given function:
\[ y = x^{6x} \]
Find:
\[ y'(x) = \text{[Input box for solution]} \]
**Solution Outline**:
To differentiate the function \( y = x^{6x} \), we can apply logarithmic differentiation as follows:
1. Take the natural logarithm of both sides:
\[ \ln y = \ln(x^{6x}) \]
2. Use the property of logarithms to simplify the right side:
\[ \ln y = 6x \ln x \]
3. Differentiate both sides with respect to \(x\):
\[
\frac{d}{dx}(\ln y) = \frac{d}{dx}(6x \ln x)
\]
4. Use the chain rule on the left side:
\[
\frac{1}{y} \cdot \frac{dy}{dx} = 6 \ln x + \frac{6x}{x}
\]
5. Simplify the right side:
\[
\frac{1}{y} \cdot \frac{dy}{dx} = 6 \ln x + 6
\]
6. Solve for \( \frac{dy}{dx} \):
\[
\frac{dy}{dx} = y(6 \ln x + 6)
\]
7. Substitute back \(y = x^{6x}\):
\[
\frac{dy}{dx} = x^{6x}(6 \ln x + 6)
\]
So the derivative is:
\[ y'(x) = x^{6x}(6 \ln x + 6) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5c58a909-3640-4198-b28f-fcaa04bb7670%2Fbc402b11-64fa-470d-91d5-a872995f3702%2F7c2l9g_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**: Use logarithmic differentiation to find the derivative of the function.
Given function:
\[ y = x^{6x} \]
Find:
\[ y'(x) = \text{[Input box for solution]} \]
**Solution Outline**:
To differentiate the function \( y = x^{6x} \), we can apply logarithmic differentiation as follows:
1. Take the natural logarithm of both sides:
\[ \ln y = \ln(x^{6x}) \]
2. Use the property of logarithms to simplify the right side:
\[ \ln y = 6x \ln x \]
3. Differentiate both sides with respect to \(x\):
\[
\frac{d}{dx}(\ln y) = \frac{d}{dx}(6x \ln x)
\]
4. Use the chain rule on the left side:
\[
\frac{1}{y} \cdot \frac{dy}{dx} = 6 \ln x + \frac{6x}{x}
\]
5. Simplify the right side:
\[
\frac{1}{y} \cdot \frac{dy}{dx} = 6 \ln x + 6
\]
6. Solve for \( \frac{dy}{dx} \):
\[
\frac{dy}{dx} = y(6 \ln x + 6)
\]
7. Substitute back \(y = x^{6x}\):
\[
\frac{dy}{dx} = x^{6x}(6 \ln x + 6)
\]
So the derivative is:
\[ y'(x) = x^{6x}(6 \ln x + 6) \]
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