derivative of-arctan(x/y)

解

\frac{d}{d x} (- arctan (\frac{x}{y}))

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

解答ステップ

\frac{d}{d x} (- arctan (\frac{x}{y}))

定数を除去:

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

= - \frac{d}{d x} (arctan (\frac{x}{y}))

連鎖法則を適用:

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

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

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

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

簡素化

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

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