Question

The derivative of sin x w.r.t. cos x is
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Answer 1

Let,

$\longrightarrow u=\sin x$

Differentiating wrt $x,$

$\longrightarrow\dfrac{du}{dx}=\dfrac{d}{dx}(\sin x)$

$\longrightarrow\dfrac{du}{dx}=\cos x\quad\quad\dots(1)$

Let,

$\longrightarrow v=\cos x$

Differentiating wrt $x,$

$\longrightarrow\dfrac{dv}{dx}=\dfrac{d}{dx}(\cos x)$

$\longrightarrow\dfrac{dv}{dx}=-\sin x\quad\quad\dots(2)$

Dividing (1) by (2),

$\longrightarrow\dfrac{\left(\dfrac{du}{dx}\right)}{\left(\dfrac{dv}{dx}\right)}=\dfrac{\cos x}{-\sin x}$

$\longrightarrow\underline{\underline{\dfrac{du}{dv}=-\cot x}}$

$\longrightarrow\underline{\underline{\dfrac{d(\sin x)}{d(\cos x)}=-\cot x}}$

This is the derivative of $\sin x$ wrt $\cos x.$