Write the derivative of sinx with respect to cosx

Let $u=\sin x$ and $v=\cos x$

$\Rightarrow \frac{d u}{d x}=\cos x$ and $\frac{d v}{d x}=-\sin x$

$\therefore \frac{\frac{d u}{d x}}{\frac{d v}{d x}}=\frac{\cos x}{-\sin x}$

$\Rightarrow \frac{d u}{d v}=-\cot x$