Calculate the derivative of y with respect to x. Express derivative in terms of sin (y³) (Express numbers in exact form. Use symbolic notation and fractions where n e4xy M dy dx Incorrect 11 - 5u(exy) (4y³ cos (14) – 5x(e5xy))
Calculate the derivative of y with respect to x. Express derivative in terms of sin (y³) (Express numbers in exact form. Use symbolic notation and fractions where n e4xy M dy dx Incorrect 11 - 5u(exy) (4y³ cos (14) – 5x(e5xy))
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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![### Calculating the Derivative of y with respect to x
**Problem Statement:**
Calculate the derivative of \( y \) with respect to \( x \). Express the derivative in terms of \( x \) and \( y \).
Given the equation:
\[ e^{4xy} = \sin(y^5) \]
**Requirements:**
- Express numbers in exact form.
- Use symbolic notation and fractions where needed.
---
**Solution:**
The incorrect attempt at a solution provided is:
\[ \frac{dy}{dx} = \frac{5y(e^{5xy})}{(4y^3 \cos(y^4) - 5xe^{5xy})} \]
The provided answer is marked as **Incorrect**.
##### Note:
To solve this problem correctly, apply implicit differentiation.
Remember that when differentiating both sides of the equation with respect to \( x \), you'll need to use the chain rule and product rule accordingly.
For \( e^{4xy} \):
\[ \frac{d(e^{4xy})}{dx} = e^{4xy} \cdot \frac{d(4xy)}{dx} = e^{4xy} \cdot (4y + 4x \frac{dy}{dx}) \]
For \( \sin(y^5) \):
\[ \frac{d(\sin(y^5))}{dx} = \cos(y^5) \cdot \frac{d(y^5)}{dx} = \cos(y^5) \cdot 5y^4 \cdot \frac{dy}{dx} \]
Setting these equal gives:
\[ e^{4xy} \cdot (4y + 4x \frac{dy}{dx}) = \cos(y^5) \cdot 5y^4 \cdot \frac{dy}{dx} \]
Isolating \(\frac{dy}{dx}\) will lead to the correct solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F575d721b-26eb-4db6-af93-50c207cd3fab%2F2e937ad4-b13e-4669-9502-8a497252ae09%2Fq1s85v9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating the Derivative of y with respect to x
**Problem Statement:**
Calculate the derivative of \( y \) with respect to \( x \). Express the derivative in terms of \( x \) and \( y \).
Given the equation:
\[ e^{4xy} = \sin(y^5) \]
**Requirements:**
- Express numbers in exact form.
- Use symbolic notation and fractions where needed.
---
**Solution:**
The incorrect attempt at a solution provided is:
\[ \frac{dy}{dx} = \frac{5y(e^{5xy})}{(4y^3 \cos(y^4) - 5xe^{5xy})} \]
The provided answer is marked as **Incorrect**.
##### Note:
To solve this problem correctly, apply implicit differentiation.
Remember that when differentiating both sides of the equation with respect to \( x \), you'll need to use the chain rule and product rule accordingly.
For \( e^{4xy} \):
\[ \frac{d(e^{4xy})}{dx} = e^{4xy} \cdot \frac{d(4xy)}{dx} = e^{4xy} \cdot (4y + 4x \frac{dy}{dx}) \]
For \( \sin(y^5) \):
\[ \frac{d(\sin(y^5))}{dx} = \cos(y^5) \cdot \frac{d(y^5)}{dx} = \cos(y^5) \cdot 5y^4 \cdot \frac{dy}{dx} \]
Setting these equal gives:
\[ e^{4xy} \cdot (4y + 4x \frac{dy}{dx}) = \cos(y^5) \cdot 5y^4 \cdot \frac{dy}{dx} \]
Isolating \(\frac{dy}{dx}\) will lead to the correct solution.
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