arctan(y/x)の偏微分

${\tan }^{-1}\left(\frac{y}{x}\right)$の偏微分

$$\begin{array}{rcl}\theta & =& {\tan }^{-1}\left(\frac{y}{x}\right)\end{array}$$

${\tan }^{-1}\left(\frac{y}{x}\right)$の$x$での偏微分

$$\begin{array}{rcl}\frac{\partial }{\partial x}{\tan }^{-1}\left(\frac{y}{x}\right)& =& \{\frac{\partial }{\partial u}{\tan }^{-1}\left(u\right)\}\frac{\partial u}{\partial x}\dots u=\frac{y}{x}\\ & =& \frac{1}{1+u^2}\frac{\partial }{\partial x}\frac{y}{x}\dots \frac{\partial }{\partial u}{\tan }^{-1}\left(u\right)=\frac{1}{1+u^2}\\ & =& \frac{1}{1+\frac{y^2}{x^2}}y\frac{\partial }{\partial x}{x}^{-1}\\ & =& \frac{1}{\frac{x^2+y^2}{x^2}}y(-x^{-2})\\ & =& \frac{x^2}{x^2+y^2}y(-x^{-2})\\ & =& \frac{-y}{x^2+y^2}\end{array}$$

${\tan }^{-1}\left(\frac{y}{x}\right)$の$y$での偏微分

$$\begin{array}{r}\\ \frac{\partial }{\partial y}{\tan }^{-1}\left(\frac{y}{x}\right)& =& \{\frac{\partial }{\partial u}{\tan }^{-1}\left(u\right)\}\frac{\partial u}{\partial y}\dots u=\frac{y}{x}\\ & =& \frac{1}{1+u^2}\frac{\partial }{\partial y}\frac{y}{x}\dots \frac{\partial }{\partial u}{\tan }^{-1}\left(u\right)=\frac{1}{1+u^2}\\ & =& \frac{1}{1+\frac{y^2}{x^2}}{x}^{-1}\frac{\partial }{\partial y}y\\ & =& \frac{1}{\frac{x^2+y^2}{x^2}}{x}^{-1}\cdot 1\\ & =& \frac{x^2}{x^2+y^2}{x}^{-1}\\ & =& \frac{x}{x^2+y^2}\end{array}$$