Answer
$$\frac{dy}{dx}=\frac{2\sqrt{xy}-y}{x + 8\sqrt{xy}}$$
Work Step by Step
Step-1: Differentiate the following equation with respect to $x$,
$$\sqrt{xy}=x-4y$$, we get,
$$\frac{1}{2\sqrt{xy}}\cdot\bigg(y+x\frac{dy}{dx}\bigg)=1-4\frac{dy}{dx}$$
Step-2: Isolate $\frac{dy}{dx}$ terms together,
$$\frac{dy}{dx}\bigg( \frac{x}{2\sqrt{xy}}+4 \bigg)=1-\frac{y}{2\sqrt{xy}}$$
$$\implies \frac{dy}{dx}=\frac{2\sqrt{xy}-y}{x + 8\sqrt{xy}}$$