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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![**Logarithmic Differentiation Example**
In this example, we will use logarithmic differentiation to find the derivative \( \frac{dy}{dx} \) of the function \( y \).
Given:
\[ y = x^{\frac{6}{x}}, \quad x > 0 \]
To find \( \frac{dy}{dx} \), we use logarithmic differentiation. The derived formula is:
\[
\frac{dy}{dx} = \left( x^{x-6} \right) \left( \ln{x} + \frac{x - 6}{x} \right)
\]
However, note that there is an incorrect notation at the end of the boxed equation indicated by a red cross mark. The detailed steps for logarithmic differentiation are not shown here, but it is important to double-check your calculations and ensure all notation is accurate. Understanding each component, such as natural logarithms and their properties during differentiation, is essential for correct results.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff9a0b810-778a-49b1-a253-be422f6ca6ba%2F7729eedd-c45d-449f-8db7-221cd9122e09%2Ffnb6ckc_processed.png&w=3840&q=75)
Transcribed Image Text:**Logarithmic Differentiation Example**
In this example, we will use logarithmic differentiation to find the derivative \( \frac{dy}{dx} \) of the function \( y \).
Given:
\[ y = x^{\frac{6}{x}}, \quad x > 0 \]
To find \( \frac{dy}{dx} \), we use logarithmic differentiation. The derived formula is:
\[
\frac{dy}{dx} = \left( x^{x-6} \right) \left( \ln{x} + \frac{x - 6}{x} \right)
\]
However, note that there is an incorrect notation at the end of the boxed equation indicated by a red cross mark. The detailed steps for logarithmic differentiation are not shown here, but it is important to double-check your calculations and ensure all notation is accurate. Understanding each component, such as natural logarithms and their properties during differentiation, is essential for correct results.
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