Question

Find the second order derivatives of $\sin \left(\log x\right)$

Answer 1

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