Question

If y = log (sin x), prove that \[\frac{d^3 y}{d x^3} = 2 \cos \ x \ {cosec}^3 x\] ?

Answer 1

\[y = \log\left( \sin x \right)\]
\[\text { Differentiating w . r . t . x, we get }\]
\[\frac{d y}{d x} = \frac{1}{\sin x} \times \cos x = \cot x\]
\[\text { Differentiating again w . r . t . x, we get }\]
\[\frac{d^2 y}{d x^2} = - {cosec}^2 x\]
\[\text { Differentiating again w . r . t . x, we get }\]
\[\frac{d^3 y}{d x^3} = - 2 \ cosec \ x \times \left( - cosec \ x \cot x \right)\]
\[ = 2\cot x {cosec}^2 x = 2\cos \ x \ {cosec}^3 x\]

Hence proved.