Answer
$$\frac{d}{dx}\tan(xy)= (y+x\frac{dy}{dx})\sec^2 (xy).$$
Work Step by Step
Recall the product rule: $(uv)'=u'v+uv'$
Recall that $(\tan x)'=\sec^2 x$.
Using the chain rule, we have
$$\frac{d}{dx}\tan(xy)= (y+x\frac{dy}{dx})\sec^2 (xy).$$