Find the given derivative. √²-1) 6 Dx 4x 3 X I Dx 4x 1 6 + 413 =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Title: Calculating the Derivative of a Function**
**Objective:**
Find the derivative of the given mathematical expression.
**Expression:**
Given:
\[ D_x \left( 4x^{-\frac{1}{3}} + \frac{6}{x^{\frac{4}{3}}} \right) \]
**Task:**
Calculate the derivative of the expression:
\[ D_x \left( 4x^{-\frac{1}{3}} + \frac{6}{x^{\frac{4}{3}}} \right) = \, \text{[Solution]} \]
**Explanation:**
The expression involves finding the derivative of two terms:
1. \( 4x^{-\frac{1}{3}} \)
2. \( \frac{6}{x^{\frac{4}{3}}} \) which can also be written as \( 6x^{-\frac{4}{3}} \)
Apply the power rule for differentiation, which states: If \( f(x) = ax^n \), then \( f'(x) = anx^{n-1} \).
**Notes:**
- Evaluate each term separately.
- Simplify the expressions to make the derivative calculation straightforward.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa3f191a8-a1b2-42ed-ac9e-32d60645d728%2F024c99f9-d23b-4efa-b69f-187042eb431f%2Fikrj02_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Calculating the Derivative of a Function**
**Objective:**
Find the derivative of the given mathematical expression.
**Expression:**
Given:
\[ D_x \left( 4x^{-\frac{1}{3}} + \frac{6}{x^{\frac{4}{3}}} \right) \]
**Task:**
Calculate the derivative of the expression:
\[ D_x \left( 4x^{-\frac{1}{3}} + \frac{6}{x^{\frac{4}{3}}} \right) = \, \text{[Solution]} \]
**Explanation:**
The expression involves finding the derivative of two terms:
1. \( 4x^{-\frac{1}{3}} \)
2. \( \frac{6}{x^{\frac{4}{3}}} \) which can also be written as \( 6x^{-\frac{4}{3}} \)
Apply the power rule for differentiation, which states: If \( f(x) = ax^n \), then \( f'(x) = anx^{n-1} \).
**Notes:**
- Evaluate each term separately.
- Simplify the expressions to make the derivative calculation straightforward.
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