What is the implicit derivative of #y-30=e^y-x^2+ye^x #?

1 Answer
Dec 30, 2015

#dy/dx=(2x-ye^x)/(e^y+e^x-1)#

Explanation:

The only tricky part of implicit differentiation is remembering to use the chain rule with anything involving #y#.

#color(red)y-color(blue)30=color(green)(e^y)-color(orange)(x^2)+color(maroon)(ye^x#

#color(red)(d/dx(y)=dy/dx#

#color(blue)(d/dx(30)=0#

#color(green)(d/dx(e^y)=e^y*dy/dx#

#color(orange)(d/dx(x^2)=2x#

Product rule coming up:

#color(maroon)(d/dx(ye^x)=dy/dx*e^x+y*d/dx(e^x)=dy/dx*e^x+ye^x#

Combine the derivatives.

#dy/dx=dy/dx*e^y-2x+dy/dx*e^x+ye^x#

Isolate the #dy/dx# terms and solve.

#2x-ye^x=dy/dx(e^y+e^x-1)#

#dy/dx=(2x-ye^x)/(e^y+e^x-1)#