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
![### Part d.
**True or False:** \(\frac{d}{dx} \frac{1}{x^3} = \frac{1}{3x^2}\).
---
**Explanation:**
This question is asking whether the derivative of \(\frac{1}{x^3}\) with respect to \(x\) is equal to \(\frac{1}{3x^2}\). To verify this, let's calculate the derivative:
1. Rewrite the function \(\frac{1}{x^3}\) as \(x^{-3}\).
2. Use the power rule for differentiation, which states that \(\frac{d}{dx} x^n = nx^{n-1}\), to find the derivative of \(x^{-3}\):
\[
\frac{d}{dx} x^{-3} = -3x^{-4}.
\]
3. Rewrite \(-3x^{-4}\) as \(\frac{-3}{x^4}\).
We see that the result is \(\frac{-3}{x^4}\), not \(\frac{1}{3x^2}\).
Therefore, the statement is **False**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9a5f5a5-4eae-476c-bcab-4a001ac27552%2F97b4d2f4-f5ee-445b-ba48-4be497318f6f%2F51a03uv_processed.png&w=3840&q=75)
Transcribed Image Text:### Part d.
**True or False:** \(\frac{d}{dx} \frac{1}{x^3} = \frac{1}{3x^2}\).
---
**Explanation:**
This question is asking whether the derivative of \(\frac{1}{x^3}\) with respect to \(x\) is equal to \(\frac{1}{3x^2}\). To verify this, let's calculate the derivative:
1. Rewrite the function \(\frac{1}{x^3}\) as \(x^{-3}\).
2. Use the power rule for differentiation, which states that \(\frac{d}{dx} x^n = nx^{n-1}\), to find the derivative of \(x^{-3}\):
\[
\frac{d}{dx} x^{-3} = -3x^{-4}.
\]
3. Rewrite \(-3x^{-4}\) as \(\frac{-3}{x^4}\).
We see that the result is \(\frac{-3}{x^4}\), not \(\frac{1}{3x^2}\).
Therefore, the statement is **False**.
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