Find the derivative of the funct = In (³√/7x - 3) y' = y =

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...
Question
**Problem Statement**

Find the derivative of the function.

\[ y = \ln\left(\sqrt[9]{7x - 3}\right) \]

**Solution**

To find the derivative \( y' \), we need to use the chain rule and logarithmic differentiation.

Let's rewrite the function:

\[ y = \ln\left((7x - 3)^{1/9}\right) \]

Using the logarithmic identity \(\ln(a^b) = b \cdot \ln(a)\), we can simplify:

\[ y = \frac{1}{9} \ln(7x - 3) \]

Now, find the derivative \( y' \):

1. Use the constant multiple rule: if \( y = c \cdot f(x) \), then \( y' = c \cdot f'(x) \).
2. The derivative of \(\ln(7x - 3)\) is \(\frac{1}{7x - 3} \cdot (7)\) using the chain rule.

So,

\[ y' = \frac{1}{9} \cdot \frac{1}{7x - 3} \cdot 7 \]

\[ y' = \frac{7}{9(7x - 3)} \]

**Answer**

\[ y' = \frac{7}{9(7x - 3)} \]
Transcribed Image Text:**Problem Statement** Find the derivative of the function. \[ y = \ln\left(\sqrt[9]{7x - 3}\right) \] **Solution** To find the derivative \( y' \), we need to use the chain rule and logarithmic differentiation. Let's rewrite the function: \[ y = \ln\left((7x - 3)^{1/9}\right) \] Using the logarithmic identity \(\ln(a^b) = b \cdot \ln(a)\), we can simplify: \[ y = \frac{1}{9} \ln(7x - 3) \] Now, find the derivative \( y' \): 1. Use the constant multiple rule: if \( y = c \cdot f(x) \), then \( y' = c \cdot f'(x) \). 2. The derivative of \(\ln(7x - 3)\) is \(\frac{1}{7x - 3} \cdot (7)\) using the chain rule. So, \[ y' = \frac{1}{9} \cdot \frac{1}{7x - 3} \cdot 7 \] \[ y' = \frac{7}{9(7x - 3)} \] **Answer** \[ y' = \frac{7}{9(7x - 3)} \]
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