Question

How do you implicitly differentiate `-1=yx+(x-y-e^y)/(y+x)`?

Answer 1

It's simple.

Explanation:

`d(-1) = 0`
`d(xy)=dx y + x dy`
`d(y+x)^(-1) = -1/(y+x)^2 (dx + dy)`
`d(x-y-e^y) = dx - dy - dy e^y`
so
`0 = y dx+ x dy -(x-y - e^y)1/(y+x)^2 (dx + dy)+1/(y+x) (dx - dy - dy e^y)`
You can continue and put all the dx and dy terms together and divide by dx. But that's going beyond what is asked, which is the differential, not the derivative.
`dy(x+(x-y-e^y)/(y+x)^2-(1+e^y)/(y+x)) = dx(-y -(x-y-e^y)/(y+x)^2 + 1/(y+x))`
This is only useful if you can separate variables.