Question

How do you implicitly differentiate `7=1-e^y/(xy)`?

Answer 1

I found: `(dy)/(dx)=y/(x(y-1))`

Explanation:

We need to remember that `y` will be itself a function of `x` so we need to derive it accordingly.
For example, if you have `y^2` the derivative will be:
`2y*(dy)/(dx)`
to consider its dependence from `x`.
In our case we have:
`0=0-(xye^y*(dy)/(dx)-e^y(y+x(dy)/(dx)))/(xy)^2`
rearranging:
`(dy)/(dx)[xye^y-xe^y]=ye^y`
`(dy)/(dx)=(ye^y)/(xe^y(y-1))=y/(x(y-1))`