Find the second - order derivatives of the function ^ -1 x - Vidyadip

Question

Find the second $-$ order derivatives of the function $\tan^{-1}x$

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Answer

Let $y = \tan^{-1}x \ \therefore \frac{{dy}}{{dx}} = \frac{1}{{1 + {x^2}}}$
$\Rightarrow \frac{{{d^2}y}}{{d{x^2}}} = \frac{d}{{dx}}\left( {\frac{1}{{1 + {x^2}}}} \right)$
$= \frac{{\left( {1 + {x^2}} \right)\frac{d}{{dx}}\left( 1 \right) - 1\frac{d}{{dx}}\left( {1 + {x^2}} \right)}}{{{{\left( {1 + {x^2}} \right)}^2}}}$
$= \frac{{\left( {1 + {x^2}} \right) \times 0 - 2x}}{{{{\left( {1 + {x^2}} \right)}^2}}} = \frac{{ - 2x}}{{{{\left( {1 + {x^2}} \right)}^2}}}$

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