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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4.1 #11
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![**Derivative Problem**
Find the given derivative.
\[
D_x \left( 5x^{-\frac{1}{3}} + \frac{6}{x^{\frac{5}{3}}} \right)
\]
\[
D_x \left( 5x^{-\frac{1}{3}} + \frac{6}{x^{\frac{5}{3}}} \right) = \boxed{}
\]
**Explanation:**
The problem requires finding the derivative of the expression with respect to \(x\). The expression consists of two terms:
1. \(5x^{-\frac{1}{3}}\) - a power of \(x\)
2. \(\frac{6}{x^{\frac{5}{3}}}\) - which can be rewritten as \(6x^{-\frac{5}{3}}\).
Apply the power rule for derivatives, \(\frac{d}{dx} x^n = nx^{n-1}\), to find the derivatives of each term separately and combine them.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F63591bc8-75d6-42dc-963a-0ef4cf7b290a%2Fbe83dab1-27d1-40ac-b9b8-89c144d61041%2F7n904a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Derivative Problem**
Find the given derivative.
\[
D_x \left( 5x^{-\frac{1}{3}} + \frac{6}{x^{\frac{5}{3}}} \right)
\]
\[
D_x \left( 5x^{-\frac{1}{3}} + \frac{6}{x^{\frac{5}{3}}} \right) = \boxed{}
\]
**Explanation:**
The problem requires finding the derivative of the expression with respect to \(x\). The expression consists of two terms:
1. \(5x^{-\frac{1}{3}}\) - a power of \(x\)
2. \(\frac{6}{x^{\frac{5}{3}}}\) - which can be rewritten as \(6x^{-\frac{5}{3}}\).
Apply the power rule for derivatives, \(\frac{d}{dx} x^n = nx^{n-1}\), to find the derivatives of each term separately and combine them.
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