Find the second order derivatives of the function given in Exerci… - Vidyadip

Question

Find the second order derivatives of the function given in Exercise:
$\sin(\log\text{x})$

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Answer

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

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