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
4.1 #8
![Title: Calculating the Derivative of a Given Function
**Objective:**
Find the derivative of the given function.
**Function:**
\[ y = \frac{-6}{2\sqrt{x}} \]
**Task:**
Calculate \(\frac{dy}{dx}\).
**Instructions:**
1. Simplify the function, if necessary.
2. Use the appropriate differentiation rules to find the derivative.
**Note:**
- \( \sqrt{x} \) can be rewritten as \( x^{1/2} \).
- Apply the power rule for differentiation, which states that \(\frac{d}{dx}(x^n) = nx^{n-1}\).
**Solution Steps:**
1. Rewrite the function:
\[ y = \frac{-6}{2} \cdot x^{-1/2} = -3x^{-1/2} \]
2. Differentiate using the power rule:
\[ \frac{dy}{dx} = -3 \cdot \left(-\frac{1}{2}\right) x^{-3/2} \]
3. Simplify:
\[ \frac{dy}{dx} = \frac{3}{2} x^{-3/2} \]
**Conclusion:**
The derivative of the function is:
\[ \frac{dy}{dx} = \frac{3}{2} x^{-3/2} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F63591bc8-75d6-42dc-963a-0ef4cf7b290a%2Fa7de8452-6e0d-479e-b6c9-6957eacf7c39%2Fx1gtfmd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Title: Calculating the Derivative of a Given Function
**Objective:**
Find the derivative of the given function.
**Function:**
\[ y = \frac{-6}{2\sqrt{x}} \]
**Task:**
Calculate \(\frac{dy}{dx}\).
**Instructions:**
1. Simplify the function, if necessary.
2. Use the appropriate differentiation rules to find the derivative.
**Note:**
- \( \sqrt{x} \) can be rewritten as \( x^{1/2} \).
- Apply the power rule for differentiation, which states that \(\frac{d}{dx}(x^n) = nx^{n-1}\).
**Solution Steps:**
1. Rewrite the function:
\[ y = \frac{-6}{2} \cdot x^{-1/2} = -3x^{-1/2} \]
2. Differentiate using the power rule:
\[ \frac{dy}{dx} = -3 \cdot \left(-\frac{1}{2}\right) x^{-3/2} \]
3. Simplify:
\[ \frac{dy}{dx} = \frac{3}{2} x^{-3/2} \]
**Conclusion:**
The derivative of the function is:
\[ \frac{dy}{dx} = \frac{3}{2} x^{-3/2} \]
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