Question

Find \(\rm \frac {d^2 \cot x}{dx^2}\)
1. 2cosec² x cot x
2. -cosec² x cot x
3. cosec³ x
4. None of these
5. cosec x

Answer 1

Correct Answer - Option 1 : 2cosec² x cot x

Concept:

\(\rm \frac {d^2 f(x)}{dx^2} = \frac {d }{ dx} \;[\frac {d }{ dx} f(x)]\)

\(\rm \frac {d }{ dx} \cot x = - \;cosec^2\; x\)

\(\rm \frac {d}{dx} coesc \;x = -cosec \;x \cot x \)

 

Calculations:

Consider, \(\rm \frac {d^2 \cot x}{dx^2} = \frac {d }{ dx} \;[\frac {d }{ dx} \cot x]\)

⇒\(\rm \frac {d^2 \cot x}{dx^2} = \frac {d }{ dx}( -cosec^2\;x)\)

⇒\(\rm \dfrac {d^2 \cot x}{dx^2} = - 2 \;cosec\; x (-cosec \;x \cot x)\)

⇒\(\rm \dfrac {d^2 \cot x}{dx^2} = 2 \;cosec^2\; x \cot x\)

Hence, \(\rm \dfrac {d^2 \cot x}{dx^2} = 2 \;cosec^2\; x \cot x\)