\(\tan^{-1}{\left(\frac{y}{x}\right)}\)の偏微分
$$\begin{eqnarray}
\theta&=&\tan^{-1}{\left(\frac{y}{x}\right)}
\end{eqnarray}$$
\(\tan^{-1}{\left(\frac{y}{x}\right)}\)の\(x\)での偏微分
$$\begin{eqnarray}
\frac{\partial }{\partial x}\tan^{-1}{\left(\frac{y}{x}\right)}
&=&\left\{\frac{\partial }{\partial u}\tan^{-1}{\left(u\right)}\right\}\frac{\partial u}{\partial x}
\;\ldots\;u=\frac{y}{x}
\\&=&\frac{1}{1+u^2}\frac{\partial }{\partial x}\frac{y}{x}
\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/07/arctanx.html}{\frac{\partial }{\partial u}\tan^{-1}{\left(u\right)}=\frac{1}{1+u^2}}
\\&=&\frac{1}{1+\frac{y^2}{x^2}}y\frac{\partial }{\partial x}x^{-1}
\\&=&\frac{1}{\frac{x^2+y^2}{x^2}}y\left(-x^{-2}\right)
\\&=&\frac{x^2}{x^2+y^2}y\left(-x^{-2}\right)
\\&=&\frac{-y}{x^2+y^2}
\end{eqnarray}$$
\(\tan^{-1}{\left(\frac{y}{x}\right)}\)の\(y\)での偏微分
$$\begin{eqnarray}
\\\frac{\partial }{\partial y}\tan^{-1}{\left(\frac{y}{x}\right)}
&=&\left\{\frac{\partial }{\partial u}\tan^{-1}{\left(u\right)}\right\}\frac{\partial u}{\partial y}
\;\ldots\;u=\frac{y}{x}
\\&=&\frac{1}{1+u^2}\frac{\partial }{\partial y}\frac{y}{x}
\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/07/arctanx.html}{\frac{\partial }{\partial u}\tan^{-1}{\left(u\right)}=\frac{1}{1+u^2}}
\\&=&\frac{1}{1+\frac{y^2}{x^2}}x^{-1}\frac{\partial }{\partial y}y
\\&=&\frac{1}{\frac{x^2+y^2}{x^2}}x^{-1}\cdot1
\\&=&\frac{x^2}{x^2+y^2}x^{-1}
\\&=&\frac{x}{x^2+y^2}
\end{eqnarray}$$
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