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:**
Differentiate the following function:
\[ f(t) = \sqrt[6]{t} - \frac{1}{\sqrt[6]{t}} \]
\[ f'(t) = \boxed{ \ } \]
**Instructions:**
To solve this problem, apply the derivative rules including the power rule and the chain rule, as required for each term in the function \( f(t) \).
Be careful to rewrite the terms in a form that makes differentiation simpler, such as changing radicals to fractional exponents.
**Step-by-Step Explanation:**
1. Rewrite the function using fractional exponents:
\[ f(t) = t^{1/6} - t^{-1/6} \]
2. Differentiate each term separately using the power rule:
\[ f'(t) = \frac{d}{dt}\left(t^{1/6}\right) - \frac{d}{dt}\left(t^{-1/6}\right) \]
Using the power rule \(\frac{d}{dt}(t^n) = nt^{n-1}\), we get:
\[ f'(t) = \frac{1}{6}t^{1/6 - 1} - \left(-\frac{1}{6}t^{-1/6 - 1}\right) \]
Simplify the exponents:
\[ f'(t) = \frac{1}{6}t^{-5/6} + \frac{1}{6}t^{-7/6} \]
Therefore, return this simplified derivative in the boxed area.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F05373f72-ba25-4c05-8ae3-aa7b15d90ca7%2F9c9b4200-fcc6-4544-804d-08c822e818b8%2Fgh8ekp_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Differentiate the following function:
\[ f(t) = \sqrt[6]{t} - \frac{1}{\sqrt[6]{t}} \]
\[ f'(t) = \boxed{ \ } \]
**Instructions:**
To solve this problem, apply the derivative rules including the power rule and the chain rule, as required for each term in the function \( f(t) \).
Be careful to rewrite the terms in a form that makes differentiation simpler, such as changing radicals to fractional exponents.
**Step-by-Step Explanation:**
1. Rewrite the function using fractional exponents:
\[ f(t) = t^{1/6} - t^{-1/6} \]
2. Differentiate each term separately using the power rule:
\[ f'(t) = \frac{d}{dt}\left(t^{1/6}\right) - \frac{d}{dt}\left(t^{-1/6}\right) \]
Using the power rule \(\frac{d}{dt}(t^n) = nt^{n-1}\), we get:
\[ f'(t) = \frac{1}{6}t^{1/6 - 1} - \left(-\frac{1}{6}t^{-1/6 - 1}\right) \]
Simplify the exponents:
\[ f'(t) = \frac{1}{6}t^{-5/6} + \frac{1}{6}t^{-7/6} \]
Therefore, return this simplified derivative in the boxed area.
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