# If f (x) = $\rm e^{\sqrt{ \ cotx}}$ , find f '( π /4 ) .

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If f (x) = $\rm e^{\sqrt{ \ cotx}}$ , find f '( π /4 )  .
1. -e /4
2. e
3. - e
4. 2e

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Correct Answer - Option 3 : - e

Concept:

• $\rm \frac{\mathrm{d} (x^{n})}{\mathrm{d} x} = n x^{n-1}$
• $\rm \frac{\mathrm{d} (\cot x)}{\mathrm{d} x} = -\ cosec^{2} x$

Calculation:

Let f (x) = $\rm e^{\sqrt{ \ cotx}}$

⇒ $\rm f' (x)= \frac{\mathrm{d} }{\mathrm{d} x} \left ( e^{\sqrt{\cot x}} \right ) = e^{\sqrt{\cot x}} \ \times \frac{1}{2\sqrt{\cot x}}\ \times \left ( - \ cosec^{2}x \right )$

⇒ $\rm f'(x) = - \frac{e^{\sqrt{\cot x}}}{2\sqrt{\cot x}} \ cosec^{2}x$

$\rm f'\left ( \frac{\pi}{4} \right ) = - \frac{e^{\sqrt{\cot \frac{\pi}{4}}}}{2\sqrt{\cot \frac{\pi}{4}}} \left ( \ cosec \frac{\pi}{4} \right )^{2}$

⇒ $\rm f'\left ( \frac{\pi}{4} \right ) = - \ e$ .

∴ The correct option is 3 .