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Find dy/dx if x²e²y = y³ In (3x).

Find dy/dx if x²e²y = y³ In (3x).

Calculus: Early Transcendentals
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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**Problem Statement:** Find \( \frac{dy}{dx} \) if \( x^2 e^{2y} = y^3 \ln(3x) \). **Explanation:** This problem involves finding the derivative \( \frac{dy}{dx} \) of an equation that mixes both \( x \) and \( y \) through implicit differentiation. The equation given is: - On the left-hand side: \( x^2 e^{2y} \), where \( e^{2y} \) indicates an exponential function of \( y \). - On the right-hand side: \( y^3 \ln(3x) \), which includes both a power of \( y \) and a natural logarithm involving \( x \). In solving this problem, apply implicit differentiation and the chain rule to express \( \frac{dy}{dx} \) in terms of \( x \) and \( y \).
Transcribed Image Text:**Problem Statement:** Find \( \frac{dy}{dx} \) if \( x^2 e^{2y} = y^3 \ln(3x) \). **Explanation:** This problem involves finding the derivative \( \frac{dy}{dx} \) of an equation that mixes both \( x \) and \( y \) through implicit differentiation. The equation given is: - On the left-hand side: \( x^2 e^{2y} \), where \( e^{2y} \) indicates an exponential function of \( y \). - On the right-hand side: \( y^3 \ln(3x) \), which includes both a power of \( y \) and a natural logarithm involving \( x \). In solving this problem, apply implicit differentiation and the chain rule to express \( \frac{dy}{dx} \) in terms of \( x \) and \( y \).
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