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Find the second order derivatives of the functions given in \( \sin (\log x) \)

$\begin{array}{1 1} \large\frac{\sin(\log x)}{x} \\ \large\frac{\cos(\log x)}{x} \\ -\Large\frac{[\cos(\log x)+\cos(\log x)]}{x^2} \\-\Large\frac{[\sin(\log x)+\cos(\log x)]}{x^2} \end{array} $

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Toolbox:
  • $y=f(x)$
  • $\large\frac{dy}{dx}$$=f'(x)$
  • $\large\frac{d^2y}{dx^2}=\frac{d}{dx}\big(\frac{dy}{dx}\big)$
  • $\large\frac{d}{dx}$$(\log x)=\large\frac{1}{x}$
Step 1:
$y=\sin (\log x)$
Differentiating with respect to $x$
$\large\frac{dy}{dx}=$$\cos (\log x).\large\frac{d}{dx}$$(\log x)$
$\quad\;=\cos(\log x).\large\frac{1}{x}$
$\quad\;=\large\frac{\cos(\log x)}{x}$
Step 2:
$\Large\frac{d^2y}{dx^2}=\frac{x.\Large\frac{d}{dx}[\cos(\log x)]-\cos(\log x).1}{x^2}$
$\quad\;=\Large\frac{[-x\sin (\log x).\Large\frac{1}{x}]-\cos (\log x).1]}{x^2}$
$\quad\;=-\Large\frac{[\sin(\log x)+\cos(\log x)]}{x^2}$
answered May 13, 2013 by sreemathi.v
 
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