How do you implicitly differentiate -y=xy+2sqrt(x-y^3) y=xy+2xy3?

1 Answer
Aug 18, 2017

dy/dx=-(x-y^3)^(1/2)/(x(x-y^3)^(1/2)+(1-3y^2)+(x-y^3)^(1/2)dydx=(xy3)12x(xy3)12+(13y2)+(xy3)12

Explanation:

-dy/dxdydx=xdy/dx+y+=xdydx+y+...

2sqrt(x+y^3)=2x+y3=2(x-y^3)^(1/2)2(xy3)12 derivative of that equal to

(x-y^3)^(-1/2)*(1-3y^2dy/dx)(xy3)12(13y2dydx)

with some math

get dy/dxdydx in one side and all in the other side

dy/dx=-(x-y^3)^(1/2)/(x(x-y^3)^(1/2)+(1-3y^2)+(x-y^3)^(1/2)dydx=(xy3)12x(xy3)12+(13y2)+(xy3)12

I need someone to check my answer or explain further, and I will be thankful.