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:**
Find the derivative of \( f(x) = x^5 \log_3(x + 7.5) \).
**Solution:**
\[ f'(x) = \]
Please enter the appropriate formulas to solve the derivative.
**Explanation:**
To find the derivative of the function \( f(x) = x^5 \log_3(x + 7.5) \), you would need to use the product rule and apply the chain rule for the logarithmic part.
1. **Product Rule**: The derivative of the product of two functions is given by:
\[ (uv)' = u'v + uv' \]
Let \( u = x^5 \) and \( v = \log_3(x + 7.5) \).
2. Find the derivatives of \( u \) and \( v \):
\[ u' = \frac{d}{dx}(x^5) = 5x^4 \]
To find \( v' \), first recall the change of base formula for logarithms:
\[ \log_3(x + 7.5) = \frac{\ln(x + 7.5)}{\ln(3)} \]
Now,
\[ v = \frac{\ln(x + 7.5)}{\ln(3)} \]
So,
\[ v' = \frac{1}{\ln(3)} \cdot \frac{d}{dx}(\ln(x + 7.5)) = \frac{1}{\ln(3)} \cdot \frac{1}{x + 7.5} \]
3. Applying the product rule:
\[ f'(x) = u'v + uv' = 5x^4 \log_3(x + 7.5) + x^5 \cdot \frac{1}{\ln(3)} \cdot \frac{1}{x + 7.5} \]
Thus,
\[ f'(x) = 5x^4 \log_3(x + 7.5) + \frac{x^5}{(x + 7.5) \ln(3)} \]
Please simplify further if required.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec7ca5ac-d913-4f83-b702-3cb8fd792592%2F7ca0d0d6-7db6-4611-a00b-34ff44007a64%2Fsa8fb9b_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the derivative of \( f(x) = x^5 \log_3(x + 7.5) \).
**Solution:**
\[ f'(x) = \]
Please enter the appropriate formulas to solve the derivative.
**Explanation:**
To find the derivative of the function \( f(x) = x^5 \log_3(x + 7.5) \), you would need to use the product rule and apply the chain rule for the logarithmic part.
1. **Product Rule**: The derivative of the product of two functions is given by:
\[ (uv)' = u'v + uv' \]
Let \( u = x^5 \) and \( v = \log_3(x + 7.5) \).
2. Find the derivatives of \( u \) and \( v \):
\[ u' = \frac{d}{dx}(x^5) = 5x^4 \]
To find \( v' \), first recall the change of base formula for logarithms:
\[ \log_3(x + 7.5) = \frac{\ln(x + 7.5)}{\ln(3)} \]
Now,
\[ v = \frac{\ln(x + 7.5)}{\ln(3)} \]
So,
\[ v' = \frac{1}{\ln(3)} \cdot \frac{d}{dx}(\ln(x + 7.5)) = \frac{1}{\ln(3)} \cdot \frac{1}{x + 7.5} \]
3. Applying the product rule:
\[ f'(x) = u'v + uv' = 5x^4 \log_3(x + 7.5) + x^5 \cdot \frac{1}{\ln(3)} \cdot \frac{1}{x + 7.5} \]
Thus,
\[ f'(x) = 5x^4 \log_3(x + 7.5) + \frac{x^5}{(x + 7.5) \ln(3)} \]
Please simplify further if required.
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