(\partial)/(\partial x)(z*arctan(y/x))

解

\frac{\partial}{\partial x} (z \cdot arctan (\frac{y}{x}))

- \frac{zy}{y ^{2} + x ^{2}}

解答ステップ

\frac{\partial}{\partial x} (z arctan (\frac{y}{x}))

z, y

を定数として扱う

定数を除去:

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

= z \frac{\partial}{\partial x} (arctan (\frac{y}{x}))

連鎖法則を適用:

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

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

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

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

簡素化

z \frac{1}{( \frac{y}{x} ) ^{2} + 1} (- \frac{y}{x ^{2}}) : - \frac{zy}{y ^{2} + x ^{2}}

= - \frac{zy}{y ^{2} + x ^{2}}