Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point. (If an answer is undefined, enter UNDEFINED.) 4exy - x = 0, (4,0) dy = At (4,0): dy
Find dy/dx by implicit differentiation. Then find the slope of the graph at the given point. (If an answer is undefined, enter UNDEFINED.) 4exy - x = 0, (4,0) dy = At (4,0): dy
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
![### Implicit Differentiation and Slope Calculation
**Problem Statement:**
Find \(\frac{dy}{dx}\) by implicit differentiation. Then find the slope of the graph at the given point. (If an answer is undefined, enter UNDEFINED.)
Equation: \(4e^{xy} - x = 0\)
Given Point: \((4, 0)\)
**Steps:**
1. **Implicit Differentiation:**
Differentiate \(4e^{xy} - x = 0\) implicitly with respect to \(x\):
\[
\frac{d}{dx}(4e^{xy}) - \frac{d}{dx}(x) = 0
\]
Using the chain rule for \(4e^{xy}\):
\[
4e^{xy} \left( x \frac{dy}{dx} + y \right) - 1 = 0
\]
2. **Solving for \(\frac{dy}{dx}\):**
Rearrange the equation to solve for \(\frac{dy}{dx}\):
\[
4e^{xy} \left( x \frac{dy}{dx} + y \right) - 1 = 0
\]
\[
4e^{xy} x \frac{dy}{dx} + 4e^{xy} y - 1 = 0
\]
\[
4e^{xy} x \frac{dy}{dx} = 1 - 4e^{xy} y
\]
\[
\frac{dy}{dx} = \frac{1 - 4e^{xy} y}{4e^{xy} x}
\]
3. **At Point \((4, 0)\):**
Substitute \(x = 4\) and \(y = 0\) into the differentiated expression:
\[
\frac{dy}{dx} = \frac{1 - 4e^{4 \cdot 0} \cdot 0}{4e^{4 \cdot 0} \cdot 4}
\]
Simplify the expression:
\[
\frac{dy}{dx} = \frac{1 - 0}{4 \cdot 4} = \frac{1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff9a0b810-778a-49b1-a253-be422f6ca6ba%2Fa8ef49bc-4b62-498f-998e-2023ec65e5bc%2Fyc4nxsb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Implicit Differentiation and Slope Calculation
**Problem Statement:**
Find \(\frac{dy}{dx}\) by implicit differentiation. Then find the slope of the graph at the given point. (If an answer is undefined, enter UNDEFINED.)
Equation: \(4e^{xy} - x = 0\)
Given Point: \((4, 0)\)
**Steps:**
1. **Implicit Differentiation:**
Differentiate \(4e^{xy} - x = 0\) implicitly with respect to \(x\):
\[
\frac{d}{dx}(4e^{xy}) - \frac{d}{dx}(x) = 0
\]
Using the chain rule for \(4e^{xy}\):
\[
4e^{xy} \left( x \frac{dy}{dx} + y \right) - 1 = 0
\]
2. **Solving for \(\frac{dy}{dx}\):**
Rearrange the equation to solve for \(\frac{dy}{dx}\):
\[
4e^{xy} \left( x \frac{dy}{dx} + y \right) - 1 = 0
\]
\[
4e^{xy} x \frac{dy}{dx} + 4e^{xy} y - 1 = 0
\]
\[
4e^{xy} x \frac{dy}{dx} = 1 - 4e^{xy} y
\]
\[
\frac{dy}{dx} = \frac{1 - 4e^{xy} y}{4e^{xy} x}
\]
3. **At Point \((4, 0)\):**
Substitute \(x = 4\) and \(y = 0\) into the differentiated expression:
\[
\frac{dy}{dx} = \frac{1 - 4e^{4 \cdot 0} \cdot 0}{4e^{4 \cdot 0} \cdot 4}
\]
Simplify the expression:
\[
\frac{dy}{dx} = \frac{1 - 0}{4 \cdot 4} = \frac{1
![**Find \( \frac{dy}{dx} \) by implicit differentiation. Then find the slope of the graph at the given point. (If an answer is undefined, enter UNDEFINED.)**
Given equation: \( 4e^{xy} - x = 0 \), at point \( (4, 0) \).
\[ \frac{dy}{dx} = \ \]
\[ \text{At} \ (4, 0): \ \frac{dy}{dx} = \ \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff9a0b810-778a-49b1-a253-be422f6ca6ba%2Fa8ef49bc-4b62-498f-998e-2023ec65e5bc%2Fh9l81wh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Find \( \frac{dy}{dx} \) by implicit differentiation. Then find the slope of the graph at the given point. (If an answer is undefined, enter UNDEFINED.)**
Given equation: \( 4e^{xy} - x = 0 \), at point \( (4, 0) \).
\[ \frac{dy}{dx} = \ \]
\[ \text{At} \ (4, 0): \ \frac{dy}{dx} = \ \]
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