If y= ( ) Prove that d^3y dx^3 =2 cosec^3x - Vidyadip

Question

If $\text{y}=\log(\sin\text{x})$ Prove that $\frac{\text{d}^3\text{y}}{\text{dx}^3}=2\cos\text{x}\ \text{cosec}^3\text{x}$

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Answer

Here,
$\text{y}=\log(\sin\text{x})$
Differentiating w.r.t.x, we get
$\frac{\text{dy}}{\text{dx}}=\frac{1}{\sin\text{x}}\times\cos\text{x}=\cot\text{x}$
Differentiating w.r.t.x, we get
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=-\text{cosec}^2\text{x}$
Differentiating w.r.t.x, we get
$\frac{\text{d}^3\text{y}}{\text{dx}^3}=-2\text{cosec}\ \text{x}\times(-\text{cosec}\ \text{x}\cot\text{x})$
$=2\cot\ \text{x}\ \text{cosec}^2\text{x}=2\cos\ \text{x}\ \text{cosec}^3\text{x}$

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