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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 .

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