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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![**Use the Product Rule to differentiate \( f(u) = \sqrt{u} \left( 5 - u^6 \right) \).**
To differentiate this function using the Product Rule, we consider the function as the product of two functions:
\( g(u) = \sqrt{u} \) and \( h(u) = 5 - u^6 \).
The Product Rule states that if \( f(u) = g(u) \cdot h(u) \), then the derivative \( f'(u) \) is given by:
\[
f'(u) = g'(u) \cdot h(u) + g(u) \cdot h'(u)
\]
**Step-by-Step Differentiation:**
1. Differentiate \( g(u) = \sqrt{u} \).
\[
g'(u) = \frac{d}{du}(\sqrt{u}) = \frac{1}{2\sqrt{u}}
\]
2. Differentiate \( h(u) = 5 - u^6 \).
\[
h'(u) = \frac{d}{du}(5 - u^6) = -6u^5
\]
3. Apply the Product Rule:
\[
f'(u) = \left(\frac{1}{2\sqrt{u}}\right)(5 - u^6) + (\sqrt{u})(-6u^5)
\]
Simplifying the expression will give you the final differentiated result.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2999936b-43d8-4fe4-88c6-ee0b16fa4ab4%2Fef57a5b0-9ebf-454c-8119-ceaaf8b65162%2F0hnrt6_processed.png&w=3840&q=75)
Transcribed Image Text:**Use the Product Rule to differentiate \( f(u) = \sqrt{u} \left( 5 - u^6 \right) \).**
To differentiate this function using the Product Rule, we consider the function as the product of two functions:
\( g(u) = \sqrt{u} \) and \( h(u) = 5 - u^6 \).
The Product Rule states that if \( f(u) = g(u) \cdot h(u) \), then the derivative \( f'(u) \) is given by:
\[
f'(u) = g'(u) \cdot h(u) + g(u) \cdot h'(u)
\]
**Step-by-Step Differentiation:**
1. Differentiate \( g(u) = \sqrt{u} \).
\[
g'(u) = \frac{d}{du}(\sqrt{u}) = \frac{1}{2\sqrt{u}}
\]
2. Differentiate \( h(u) = 5 - u^6 \).
\[
h'(u) = \frac{d}{du}(5 - u^6) = -6u^5
\]
3. Apply the Product Rule:
\[
f'(u) = \left(\frac{1}{2\sqrt{u}}\right)(5 - u^6) + (\sqrt{u})(-6u^5)
\]
Simplifying the expression will give you the final differentiated result.
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