Answer
$-\frac{y}{x}$
Work Step by Step
$xy= cot (xy)$
Differentiating directly with respect to x, we have
$\frac{d}{dx}(xy)= \frac{d}{dx}(cot (xy))$
Using product rule and chain rule, we get
$y+xy'= (-cosec^{2}(xy))(y+xy')$
$y+xy'= -xy'cosec^{2}(xy)-ycosec^{2}(xy)$
Adding $xy'cosec^{2}(xy)$ to both sides and substracting $y$ from both sides, we obtain
$xy'+ xy'cosec^{2}(xy)= -y-ycosec^{2}(xy)$
Taking out the factors, we have
$xy'(1+cosec^{2}(xy))= -y(1+cosec^{2}(xy))$
$⇒xy'= -y$
$⇒\frac{dy}{dx}= \frac{-y}{x}$
