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Each of the equations in Exercises 69–80 defines y as an implicit function of x. Use implicit differentiation (without solving for y first) to find dydx

(3y2+5xy2)4=2

Short Answer

Expert verified

dydx=-5y(5x+6y)

Step by step solution

01

Step 1. Given Information:

Given equation:(3y2+5xy2)4=2

We want to find dydxdefines y as an implicit function of x by use implicit differentiation.

02

Step 2. Solution: 

Differentiate both sides w.r.t. x

ddx(3y2+5xy2)4=ddx2

Using product and chain rule we get

localid="1648700033816" 4(3y2+5xy2)3ddx(3y2+5xy2)=ddx24(3y2+5xy2)3ddx(3y2)+ddx(5xy)ddx(2)=ddx24(3y2+5xy2)36ydydx+5xdydx+5y0=0(5x+6y)dydx+5y=0(5x+6y)dydx=-5ydydx=-5y(5x+6y)

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