arctan(x)の微分

${\tan }^{-1}\left(x\right)$の微分

$$\begin{array}{rcl}y& =& {\tan }^{-1}\left(x\right)\\ x& =& \tan \left(y\right)\\ \frac{\mathrm{d}x}{\mathrm{d}y}& =& \frac{\mathrm{d}}{\mathrm{d}y}\tan \left(y\right)\\ & =& 1+{\tan }^2\left(y\right)\\ \frac{\mathrm{d}y}{\mathrm{d}x}& =& \frac{1}{1+{\tan }^2\left(y\right)}\dots \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{\frac{\mathrm{d}x}{\mathrm{d}y}}\left(\mathrm{逆}\mathrm{凾}\mathrm{数}\mathrm{の}\mathrm{微}\mathrm{分}\right)\\ & =& \frac{1}{1+x^2}\dots x=\tan \left(y\right)\end{array}$$