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 .
