Differentiate x x respect to x . - Vidyadip

Question

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

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Answer

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

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