Question

If y = log (sin x), prove that $\frac{d^{3}y}{d{x}^3}=2\cos x{\mathrm{cosec}}^{3}x$.

Answer 1

Here,
$$\begin{gathered} y=\log \left(\sin x\right) \\ \mathrm{Differentiating}\mathrm{w}.\mathrm{r}.\mathrm{t}.\mathit{}x,\mathrm{we}\mathrm{get} \\ \frac{\mathit{d}y}{\mathit{d}x}=\frac{1}{\sin x}\times \cos x=\cot x \\ \mathrm{Differentiating}\mathrm{again}\mathrm{w}.\mathrm{r}.\mathrm{t}.\mathit{}x,\mathrm{we}\mathrm{get} \\ \frac{{\mathit{d}}^{2}y}{\mathit{d}{x}^2}=-{\mathrm{cosec}}^{2}x \\ \mathrm{Differentiating}\mathrm{again}\mathrm{w}.\mathrm{r}.\mathrm{t}.\mathit{}x,\mathrm{we}\mathrm{get} \\ \frac{{\mathit{d}}^{3}y}{\mathit{d}{x}^3}=-2\mathrm{cosec}x\times \left(-\mathrm{cosec}x\cot x\right) \\ =2\cot x{\mathrm{cosec}}^{2}x=2\cos x{\mathrm{cosec}}^{3}x \end{gathered}$$

Hence proved.