Find the second order derivatives of the following functions: ( ) - Vidyadip

Question

Find the second order derivatives of the following functions:$\sin(\log\text{x})$

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Answer

Let $\text{y}=\sin(\log\text{x})$
Then
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}[\sin(\log\text{x})]=\cos(\log\text{x}).\frac{\text{d}}{\text{dx}}(\log\text{x})=\frac{\cos(\log\text{x})}{\text{x}}$
$\therefore\frac{\text{d}^2\text{y}}{\text{dx}^2}=\frac{\text{d}}{\text{dx}}\Big[\frac{\cos(\log\text{x})}{\text{x}}\Big]$
$=\frac{\text{x}.\frac{\text{d}}{\text{dx}}[\cos(\log\text{x})]-\cos(\log\text{x}).\frac{\text{d}}{\text{dx}}(\text{x})}{\text{x}^2}$
$=\frac{\text{x}.\Big[-\sin(\log\text{x}).\frac{\text{d}}{\text{dx}(\log\text{x})}\Big]-\cos(\log\text{x}.1}{\text{x}^2}$
$\frac{-\text{x}\sin(\log\text{x}).\frac{1}{\text{x}}-\cos(\log\text{x})}{\text{x}^2}$
$=\frac{[-\sin(\log\text{x})+\cos(\log\text{x})]}{\text{x}^2}$

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