Question

The derivative of cos x with respect to sin x is __________________.

Answer 1

Let u(x) = cosx and v(x) = sinx.

$u\left(x\right)=\cos x$

$\Rightarrow \frac{du}{dx}=\frac{d}{dx}\cos x$

$\Rightarrow \frac{du}{dx}=-\sin x$ .....(1)

$v\left(x\right)=\sin x$

$\Rightarrow \frac{dv}{dx}=\frac{d}{dx}\sin x$

$\Rightarrow \frac{dv}{dx}=\cos x$ .....(2)

$\therefore \frac{du}{dv}=\frac{{\displaystyle \frac{du}{dx}}}{{\displaystyle \frac{dv}{dx}}}$

$\Rightarrow \frac{du}{dv}=\frac{{\displaystyle -\sin x}}{{\displaystyle \cos x}}$ [From (1) and (2)]

$\Rightarrow \frac{du}{dv}=-\tan x$

Thus, the derivative of cosx with respect to sinx is −tanx.


The derivative of cos x with respect to sin x is ___−tanx___.