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
![**Problem Statement:**
Find \(\frac{dy}{dx}\) for \(y = \frac{x^2}{\sqrt{9 - x^4}}\).
---
**Solution:**
To find \(\frac{dy}{dx}\), you need to apply the quotient rule and the chain rule.
1. **Quotient Rule:** If you have a function \(\frac{u}{v}\), the derivative is given by:
\[
\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}
\]
Here, \(u = x^2\) and \(v = \sqrt{9 - x^4}\).
2. **Find \(\frac{du}{dx}\):**
\[
\frac{du}{dx} = 2x
\]
3. **Find \(\frac{dv}{dx}\):**
- First, express \(v = (9 - x^4)^{1/2}\).
- Use the chain rule: the derivative of \((9 - x^4)^{1/2}\) is:
\[
\frac{1}{2}(9 - x^4)^{-1/2} \cdot (-4x^3) = -2x^3(9 - x^4)^{-1/2}
\]
4. **Apply the Quotient Rule:**
- Substitute back into the quotient rule formula:
\[
\frac{dy}{dx} = \frac{\sqrt{9 - x^4} \cdot 2x - x^2 \cdot \left(-2x^3(9 - x^4)^{-1/2}\right)}{(9 - x^4)}
\]
5. **Simplify:**
- Combine terms:
\[
\frac{dy}{dx} = \frac{2x\sqrt{9 - x^4} + 2x^5(9 - x^4)^{-1/2}}{9 - x^4}
\]
This expression represents the derivative of the function \(y\) with respect to \(x\). Be sure to include further simplification if necessary.
**Final Answer:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc17f3f0c-3f3b-4d84-90ac-d23f7cabcc9b%2F39aa4c94-f8b2-4655-a956-b0edbe167772%2Fqxxvakb_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find \(\frac{dy}{dx}\) for \(y = \frac{x^2}{\sqrt{9 - x^4}}\).
---
**Solution:**
To find \(\frac{dy}{dx}\), you need to apply the quotient rule and the chain rule.
1. **Quotient Rule:** If you have a function \(\frac{u}{v}\), the derivative is given by:
\[
\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}
\]
Here, \(u = x^2\) and \(v = \sqrt{9 - x^4}\).
2. **Find \(\frac{du}{dx}\):**
\[
\frac{du}{dx} = 2x
\]
3. **Find \(\frac{dv}{dx}\):**
- First, express \(v = (9 - x^4)^{1/2}\).
- Use the chain rule: the derivative of \((9 - x^4)^{1/2}\) is:
\[
\frac{1}{2}(9 - x^4)^{-1/2} \cdot (-4x^3) = -2x^3(9 - x^4)^{-1/2}
\]
4. **Apply the Quotient Rule:**
- Substitute back into the quotient rule formula:
\[
\frac{dy}{dx} = \frac{\sqrt{9 - x^4} \cdot 2x - x^2 \cdot \left(-2x^3(9 - x^4)^{-1/2}\right)}{(9 - x^4)}
\]
5. **Simplify:**
- Combine terms:
\[
\frac{dy}{dx} = \frac{2x\sqrt{9 - x^4} + 2x^5(9 - x^4)^{-1/2}}{9 - x^4}
\]
This expression represents the derivative of the function \(y\) with respect to \(x\). Be sure to include further simplification if necessary.
**Final Answer:
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