First slide

Diffrentiation

Question

$If y = e^{x} \cdot \cot x then \frac{d y}{d x} will be$

Moderate

A

$e^{x} \cot x - {cosec}^{2} x$
B

$e^{x} {cosec}^{2} x$
C

$e^{x} [\cot x - {cosec}^{2} x]$
D

$e^{x} \cot x$

Solution

$\begin{array}{l} y = e^{x} \cot x \\ \frac{d y}{d x} = e^{x} \frac{d}{d x} \cot x + \cot x \frac{d e^{x}}{d x} (∵ \frac{d u v}{d x} = \frac{u d v}{d x} + \frac{v d u}{d x}) \\ = e^{x} [- {cosec}^{2} x] + \cot x [e^{x}] (∵ \frac{d}{d x} \cot x = - {cosec}^{2} x \frac{d e^{x}}{d x} = e^{x}) \\ = - {cosec}^{2} x e^{x} + e^{x} \cot x \end{array}$

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