Compute the derivative. If f(x) and g(x) are differentiable functions such that f(2) = 6, f'(2) = 2, g(2) = -1, g'(2) = -5, compute the following derivative: (f(x)g(x)] dx x-2 O 32 O 17 O -32 O -28
Compute the derivative. If f(x) and g(x) are differentiable functions such that f(2) = 6, f'(2) = 2, g(2) = -1, g'(2) = -5, compute the following derivative: (f(x)g(x)] dx x-2 O 32 O 17 O -32 O -28
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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![## Compute the Derivative
Consider two differentiable functions, \( f(x) \) and \( g(x) \), with the given conditions:
- \( f(2) = 6 \)
- \( f'(2) = 2 \)
- \( g(2) = -1 \)
- \( g'(2) = 5 \)
### Problem
Compute the derivative of the product \( f(x)g(x) \) at \( x = 2 \).
### Solution
To solve this problem, we use the product rule for derivatives. The product rule states:
\[
\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
\]
Plugging in the given values at \( x = 2 \):
\[
f'(2)g(2) + f(2)g'(2) = (2)(-1) + (6)(5)
\]
\[
= -2 + 30
\]
\[
= 28
\]
### Answer Choices
- O 32
- O 17
- O -32
- O -28
The correct option for the derivative at \( x = 2 \) is 28. Note: There seems to be no option matching this correct result, which should be addressed or checked for potential discrepancies.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2f259b4-cd04-4715-ba74-9a8f7fc0fecb%2Fe869a69b-2063-4bde-b40d-cfe6657c34e9%2Ffmhr89_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Compute the Derivative
Consider two differentiable functions, \( f(x) \) and \( g(x) \), with the given conditions:
- \( f(2) = 6 \)
- \( f'(2) = 2 \)
- \( g(2) = -1 \)
- \( g'(2) = 5 \)
### Problem
Compute the derivative of the product \( f(x)g(x) \) at \( x = 2 \).
### Solution
To solve this problem, we use the product rule for derivatives. The product rule states:
\[
\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
\]
Plugging in the given values at \( x = 2 \):
\[
f'(2)g(2) + f(2)g'(2) = (2)(-1) + (6)(5)
\]
\[
= -2 + 30
\]
\[
= 28
\]
### Answer Choices
- O 32
- O 17
- O -32
- O -28
The correct option for the derivative at \( x = 2 \) is 28. Note: There seems to be no option matching this correct result, which should be addressed or checked for potential discrepancies.
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