How do I find dy/dx of this problem?
$\displaystyle tan (x/y) = x + y$
do you realize that everytime you differentiate a y-term you attach dy/dx to it?
so, to start you off.
$\displaystyle \tan \frac xy = x + y$
$\displaystyle \Rightarrow \sec^2 \left( \frac xy \right) \cdot \frac d{dx} \left( \frac xy \right) = 1 + \frac {dy}{dx}$
(the left hand side follows by the chain rule)
can you continue?