Question
Find the second-order derivative of the function sin(log x)
$\therefore \frac{{dy}}{{dx}} = \cos \left( {\log x} \right)\frac{d}{{dx}}\left( {\log x} \right)$
$= \cos \left( {\log x} \right).\frac{1}{x} = \frac{{\cos \left( {\log x} \right)}}{x}$
$\Rightarrow \frac{{{d^2}y}}{{d{x^2}}} = \frac{{x\frac{d}{{dx}}\cos \left( {\log x} \right) - \cos \left( {\log x} \right)\frac{d}{{dx}}x}}{{{x^2}}}$
$= \frac{{x (-sinlogx)\frac{d}{{dx}}\left( {\log x} \right) - \cos \left( {\log x} \right) \times 1}}{{{x^2}}}$
$= \frac{{ - x\sin \left( {\log x} \right)\frac{1}{x} - \cos \left( {\log x} \right)}}{{{x^2}}}$
$= \frac{{ - \left[ {\sin \left( {\log x} \right) + \cos \left( {\log x } \right)} \right]}}{{{x^2}}}$
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