Please refer to photo and show steps! Thanks! **[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 \( -\frac{x^2}{y^2} \).

Given implicit function : x3+y3=5For dydx, taking derivative of x3+y3=5 with respect to x. ⇒ddx(x3+y3)=ddx(5)⇒ddx(x3)+ddx(y3)=0 ∵ddx(u+v)=dudx+dvdx⇒3x2+3y2dydx=0 ∵ddx(xn)=nxn-1⇒3y2dydx=-3x2⇒dydx=-x2y2

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Math

Calculus

Find dy/dx by implicit differentiation. 52) x3 + y3 = 5

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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