Question

Write the derivative of $\sin x$ w.r.t. $\cos x$ is $-cotx$.

Write $1$ if true and $0$ if false.

Answer 1

Let $u=\sin x,v=\cos x$
So, we need to find $\frac{du}{dv}$
On differentiating w.r. t. 'x', we get

$\frac{du}{dx}=\frac{d}{dx}\left(\sin x\right)=\cos x$ and

$\frac{dv}{dx}=\frac{d}{dx}\left(\cos x\right)=-\sin x$

$\therefore \frac{du}{dv}=\frac{\cos x}{-\sin x}=-\cot x$