Answer to Question #137413 in Calculus for Nick

Question #137413

use implicit differentiantion to find the derivative of the fuction

cos(x^2+2y) + xe^y = 0


1
Expert's answer
2020-10-08T16:04:39-0400

cos(x2+2y)+xey=0cos(x^2+2y) + xe^y = 0


ddx[cos(x2+2y)+xey]=ddx[0]\frac{d}{dx}[cos(x^2+2y) + xe^y]=\frac{d}{dx}[0]


ddx[cos(x2+2y)+xey]=ddx[0]\frac{d}{dx}[cos(x^2+2y) + xe^y]=\frac{d}{dx}[0]


(sin(x2+2y)).ddx[x2+2y]+ddx[ey].x+ey.ddx[x]=0(-sin(x^2+2y)).\frac{d}{dx}[x^2+2y]+\frac{d}{dx}[e^y].x+e^y.\frac{d}{dx}[x]=0


(ddx[x2]+ddx[2y])sin(x2+2y)+ey.ddx[y].x+ey.1=0-(\frac{d}{dx}[x^2]+\frac{d}{dx}[2y])sin(x^2+2y)+e^y.\frac{d}{dx}[y].x+e^y.1=0


(2x+2y)sin(x2+2y)+eyyx+ey=0-(2x+2y\prime)sin(x^2+2y)+e^yy\prime x+e^y=0


(2y+2x)sin(2y+x2)+xeyy+ey=0-(2y\prime +2x)sin(2y+x^2)+xe^yy\prime +e^y=0


y=2xsin(2y+x2)ey2sin(2y+x2)xeyy\prime =\frac{2xsin(2y+x^2)-e^y}{2sin(2y+x^2)-xe^y}


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