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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Question
Explain and label each step. Include all details
![**Differentiate the following function.**
Given:
\[ u = \sqrt[5]{t^2} + 4\sqrt{t^3} \]
Find:
\[ u'(t) = \]
---
**Instructions:**
To differentiate the function \( u \), apply the power rule and the chain rule. The first term is \( \sqrt[5]{t^2} \), which can be rewritten as \( t^{2/5} \). The second term is \( 4\sqrt{t^3} \), which can be rewritten as \( 4t^{3/2} \).
Differentiate each term:
1. \( \frac{d}{dt} t^{2/5} = \frac{2}{5} t^{-3/5} \)
2. \( \frac{d}{dt} 4t^{3/2} = 4 \times \frac{3}{2} t^{1/2} = 6t^{1/2} \)
Therefore:
\[ u'(t) = \frac{2}{5} t^{-3/5} + 6t^{1/2} \]
Insert your answer in the box provided above. If the entered answer is incorrect, it will show a red cross indicating the error.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F38650a9c-63f5-423f-8b89-b03c0dad949f%2F3affe8e7-1824-4799-9ac3-a754f11e2672%2Ftr4k3vk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Differentiate the following function.**
Given:
\[ u = \sqrt[5]{t^2} + 4\sqrt{t^3} \]
Find:
\[ u'(t) = \]
---
**Instructions:**
To differentiate the function \( u \), apply the power rule and the chain rule. The first term is \( \sqrt[5]{t^2} \), which can be rewritten as \( t^{2/5} \). The second term is \( 4\sqrt{t^3} \), which can be rewritten as \( 4t^{3/2} \).
Differentiate each term:
1. \( \frac{d}{dt} t^{2/5} = \frac{2}{5} t^{-3/5} \)
2. \( \frac{d}{dt} 4t^{3/2} = 4 \times \frac{3}{2} t^{1/2} = 6t^{1/2} \)
Therefore:
\[ u'(t) = \frac{2}{5} t^{-3/5} + 6t^{1/2} \]
Insert your answer in the box provided above. If the entered answer is incorrect, it will show a red cross indicating the error.
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