If `y=log(sinx)` , prove that `(d^3y)/(dx^3)=2cosx cos e c^3x` .

Explanation

Suppose `y=\log (\sin x)`


First finding` d y / d x`



`\therefore \frac{d y}{d x}=\frac{d}{d x}(\log (\sin x)) Let, t=\sin x and y=\log t \because \frac{d y}{d x}=\frac{d y}{d t} \times \frac{d t}{d x}[ using chain rule] \frac{d y}{d x}=\cos x \times \frac{1}{t} {[\because \frac{d}{d x} \log x=\frac{1}{x} \ \frac{d}{d x}(\sin x)=\cos x] } \frac{d y}{d x}=\frac{\cos x}{\sin x}=\cot x`



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