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Q. The derivative of $sin\, x$ with respect to $ cos\text{ }x $ is

J & K CETJ & K CET 2015Continuity and Differentiability

Solution:

Let $ u(x)=\sin \,x $
and $ v(x)=\cos \,x\,\,\,\Rightarrow \,\,\frac{du}{dx}=\cos \,x $
and $ \frac{dv}{dx}=-\sin x $
Then, $ \frac{du}{dv}=\frac{du/dx}{dv/dx}\Rightarrow \frac{du}{dv}=\frac{\cos \,x}{-\sin x} $
$ \Rightarrow $ $ \frac{du}{dv}=-\cot \,x $