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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![The image contains the mathematical expression:
\[ g(x) = \log_3(5x^4 - 3x^2) \]
The task is to find the derivative \( g'(x) \).
### Explanation for an Educational Website:
To find the derivative \( g'(x) \) of the function \( g(x) = \log_3(5x^4 - 3x^2) \), we must apply the chain rule and the properties of logarithms. Here, the base of the logarithm is 3, and the expression inside the logarithm is \( 5x^4 - 3x^2 \).
**Steps:**
1. **Recall the derivative of a logarithm with a base \( a \):**
\[
\frac{d}{dx}[\log_a(u)] = \frac{1}{u \ln(a)} \cdot \frac{du}{dx}
\]
where \( u \) is a function of \( x \).
2. **Identify \( u \) in the given problem:**
\( u = 5x^4 - 3x^2 \)
3. **Find the derivative \( \frac{du}{dx} \):**
\[
\frac{d}{dx}(5x^4 - 3x^2) = 20x^3 - 6x
\]
4. **Substitute into the derivative formula:**
\[
g'(x) = \frac{1}{(5x^4 - 3x^2) \ln(3)} \cdot (20x^3 - 6x)
\]
5. **Simplify if possible:**
\[
g'(x) = \frac{20x^3 - 6x}{(5x^4 - 3x^2) \ln(3)}
\]
This formula gives the derivative of the given logarithmic function with respect to \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb6a1094-a755-4726-bad3-dc2538d25fe7%2Fa2f370c3-b21f-4af8-bed5-c73c9e75a0ed%2F5iaaqtn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains the mathematical expression:
\[ g(x) = \log_3(5x^4 - 3x^2) \]
The task is to find the derivative \( g'(x) \).
### Explanation for an Educational Website:
To find the derivative \( g'(x) \) of the function \( g(x) = \log_3(5x^4 - 3x^2) \), we must apply the chain rule and the properties of logarithms. Here, the base of the logarithm is 3, and the expression inside the logarithm is \( 5x^4 - 3x^2 \).
**Steps:**
1. **Recall the derivative of a logarithm with a base \( a \):**
\[
\frac{d}{dx}[\log_a(u)] = \frac{1}{u \ln(a)} \cdot \frac{du}{dx}
\]
where \( u \) is a function of \( x \).
2. **Identify \( u \) in the given problem:**
\( u = 5x^4 - 3x^2 \)
3. **Find the derivative \( \frac{du}{dx} \):**
\[
\frac{d}{dx}(5x^4 - 3x^2) = 20x^3 - 6x
\]
4. **Substitute into the derivative formula:**
\[
g'(x) = \frac{1}{(5x^4 - 3x^2) \ln(3)} \cdot (20x^3 - 6x)
\]
5. **Simplify if possible:**
\[
g'(x) = \frac{20x^3 - 6x}{(5x^4 - 3x^2) \ln(3)}
\]
This formula gives the derivative of the given logarithmic function with respect to \( x \).
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