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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![**[6.4] Find dy/dx by implicit differentiation.**
52) \( x^3 + y^3 = 5 \)
To find \( \frac{dy}{dx} \) using implicit differentiation, apply the derivative to both sides of the equation:
1. Differentiate \( x^3 \) with respect to \( x \) to get \( 3x^2 \).
2. Differentiate \( y^3 \) with respect to \( x \). Use the chain rule: \( 3y^2 \cdot \frac{dy}{dx} \).
3. The derivative of the constant 5 is 0.
Combining these results:
\[ 3x^2 + 3y^2 \cdot \frac{dy}{dx} = 0 \]
Next, solve for \( \frac{dy}{dx} \):
1. Subtract \( 3x^2 \) from both sides:
\[ 3y^2 \cdot \frac{dy}{dx} = -3x^2 \]
2. Divide by \( 3y^2 \):
\[ \frac{dy}{dx} = -\frac{x^2}{y^2} \]
Thus, the derivative \( \frac{dy}{dx} \) is \( -\frac{x^2}{y^2} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff3b3359e-a879-4588-bd5e-cbb2de0925ec%2F1f7dbef0-471b-416a-9181-0091f6eb9f39%2Fxuukiii_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**[6.4] Find dy/dx by implicit differentiation.**
52) \( x^3 + y^3 = 5 \)
To find \( \frac{dy}{dx} \) using implicit differentiation, apply the derivative to both sides of the equation:
1. Differentiate \( x^3 \) with respect to \( x \) to get \( 3x^2 \).
2. Differentiate \( y^3 \) with respect to \( x \). Use the chain rule: \( 3y^2 \cdot \frac{dy}{dx} \).
3. The derivative of the constant 5 is 0.
Combining these results:
\[ 3x^2 + 3y^2 \cdot \frac{dy}{dx} = 0 \]
Next, solve for \( \frac{dy}{dx} \):
1. Subtract \( 3x^2 \) from both sides:
\[ 3y^2 \cdot \frac{dy}{dx} = -3x^2 \]
2. Divide by \( 3y^2 \):
\[ \frac{dy}{dx} = -\frac{x^2}{y^2} \]
Thus, the derivative \( \frac{dy}{dx} \) is \( -\frac{x^2}{y^2} \).
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