Question

How do you implicitly differentiate `-y=xy-xe^y`?

Answer 1

`(dy)/(dx) = (y-e^y)/(xe^y-x-1)`

Explanation:

I'm finding `(dy)/(dx)`. Differentiate term by term, the derivative of y wrt x is `(dy)/(dx)` while remembering to be careful with our product rule for both terms on the right hand side.

`-(dy)/(dx) = y + x(dy)/(dx) - e^y - x(dy)/(dx)e^y`

Collect like terms

`(dy)/(dx)[xe^y-x-1] = y -e^y`

and solve:

`(dy)/(dx) = (y-e^y)/(xe^y-x-1)`