arctan(x)の微分

\(\tan^{-1}{\left(x\right)}\)の微分

$$\begin{eqnarray} y&=&\tan^{-1}{\left(x\right)} \\x&=&\tan{\left(y\right)} \\\frac{\mathrm{d}x}{\mathrm{d}y}&=&\frac{\mathrm{d}}{\mathrm{d}y}\tan{\left(y\right)} \\&=&1+\tan^2{\left(y\right)} \\\frac{\mathrm{d}y}{\mathrm{d}x}&=&\frac{1}{1+\tan^2{\left(y\right)}} \;\ldots\;\href{https://shikitenkai.blogspot.com/2021/07/blog-post.html}{\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{\frac{\mathrm{d}x}{\mathrm{d}y}}\;\left(逆凾数の微分\right)} \\&=&\frac{1}{1+x^2} \;\ldots\;\href{https://shikitenkai.blogspot.com/2021/07/tanx-2.html}{x=\tan{\left(y\right)}} \end{eqnarray}$$