Differentiate x x with respect to x. - Vidyadip

Question

Differentiate $\frac{x}{\sin x}$ with respect to $x$.

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Answer

Let, $f(x)=\frac{x}{\sin x}$
$\therefore \quad f^{\prime}(x)=\frac{d}{d x}\left(\frac{x}{\sin x}\right)$
$\begin{array}{l}=\frac{\sin x \frac{d}{d x} x-x \frac{d}{d x} \sin x}{(\sin x)^2} \\ =\frac{\sin x \cdot 1-x \cdot \cos x}{\sin ^2 x} \\ =\operatorname{cosec} x-x \cot x \operatorname{cosec} x \\ =(1-x \cot x) \cdot \operatorname{cosec} x\end{array}$

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