Question

# ( y=log (operatorname{sen} x) quad ) Proove the (frac{d^{3} y}{d x^{2}}=2 cos x cdot operatorname{cosec}^{3} x )

Solution

( begin{aligned} y &=log (sin x) frac{d y}{d x} &=frac{(cos x)}{sin x}=cot x frac{d^{2} y}{d x^{2}}=-operatorname{cosec}^{2} x Rightarrow frac{d^{3} y}{d x^{2}} &=-2 operatorname{cosec} x-(operatorname{cosec} x cot x) text { prois } &=2 operatorname{cosec}^{2} x frac{cos x}{sin x} &=2 operatorname{cosec}^{3} x cos x end{aligned} )