Question

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

Answer 1

`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)`