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 the function. *(Factor your answer completely.)*
Given function:
\[ f(x) = (x + 1)e^x \]
Find:
\[ f'(x) = \, ? \]
**Explanation:**
To find the derivative of the function \( f(x) = (x + 1)e^x \), use the product rule. The product rule states that if you have a function that is the product of two functions, \( u(x) \) and \( v(x) \), then the derivative is:
\[ (uv)' = u'v + uv' \]
Here, let:
- \( u(x) = x + 1 \), so \( u'(x) = 1 \)
- \( v(x) = e^x \), so \( v'(x) = e^x \)
Applying the product rule:
\[ f'(x) = u'v + uv' = (1)e^x + (x + 1)e^x \]
Simplify and factor completely:
\[ f'(x) = e^x + (x + 1)e^x = e^x(1 + x + 1) \]
\[ f'(x) = e^x(x + 2) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87946fbe-26a8-44e7-b719-4f36f43dcbac%2F91f3909d-7dd9-46fd-b498-aea5ae831c5b%2F2nca96_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the derivative of the function. *(Factor your answer completely.)*
Given function:
\[ f(x) = (x + 1)e^x \]
Find:
\[ f'(x) = \, ? \]
**Explanation:**
To find the derivative of the function \( f(x) = (x + 1)e^x \), use the product rule. The product rule states that if you have a function that is the product of two functions, \( u(x) \) and \( v(x) \), then the derivative is:
\[ (uv)' = u'v + uv' \]
Here, let:
- \( u(x) = x + 1 \), so \( u'(x) = 1 \)
- \( v(x) = e^x \), so \( v'(x) = e^x \)
Applying the product rule:
\[ f'(x) = u'v + uv' = (1)e^x + (x + 1)e^x \]
Simplify and factor completely:
\[ f'(x) = e^x + (x + 1)e^x = e^x(1 + x + 1) \]
\[ f'(x) = e^x(x + 2) \]
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