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Find the second order derivatives of the functions given in \( \tan^{-1} x \)

$\begin{array}{1 1} \large\frac{2x}{(1-x^2)^2}\\\large\frac{-2x}{(1+x^2)^2} \\ \large\frac{-2x}{(1+x^2)^3} \\(1+x^2)^{-1} \end{array} $

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Toolbox:
  • $y=f(x)$
  • $\large\frac{dy}{dx}$$=f'(x)$
  • $\large\frac{d^2y}{dx^2}=\frac{d}{dx}\big(\frac{dy}{dx}\big)$
  • $\large\frac{d}{dx}$$(\tan^{-1}x)=\large\frac{1}{1+x^2}$
Step 1:
$y=\tan^{-1}x$
Differentiating with respect to $x$
$\large\frac{dy}{dx}=\frac{1}{1+x^2}$
$\quad\;=(1+x^2)^{-1}$
Step 2:
$\large\frac{d^2y}{dx^2}$$=\large\frac{d}{dx}$$(1+x^2)^{-1}$
$\quad\;=(-1).(1+x^2)^{-2}.2x$
$\quad\;=\large\frac{-2x}{(1+x^2)^2}$
answered May 13, 2013 by sreemathi.v
edited Mar 8 by meena.p
 
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