(\partial)/(\partial x)(1/2 yarctan(x/y))

解

\frac{\partial}{\partial x} (\frac{1}{2} y arctan (\frac{x}{y}))

\frac{y ^{2}}{2 ( x ^{2} + y ^{2} )}

解答ステップ

\frac{\partial}{\partial x} (\frac{1}{2} y arctan (\frac{x}{y}))

y

を定数として扱う

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{1}{2} y \frac{\partial}{\partial x} (arctan (\frac{x}{y}))

連鎖法則を適用:

\frac{1}{( \frac{x}{y} ) ^{2} + 1} \frac{\partial}{\partial x} (\frac{x}{y})

= \frac{1}{( \frac{x}{y} ) ^{2} + 1} \frac{\partial}{\partial x} (\frac{x}{y})

\frac{\partial}{\partial x} (\frac{x}{y}) = \frac{1}{y}

= \frac{1}{2} y \frac{1}{( \frac{x}{y} ) ^{2} + 1} \cdot \frac{1}{y}

簡素化

\frac{1}{2} y \frac{1}{( \frac{x}{y} ) ^{2} + 1} \cdot \frac{1}{y} : \frac{y ^{2}}{2 ( x ^{2} + y ^{2} )}

= \frac{y ^{2}}{2 ( x ^{2} + y ^{2} )}