derivative of arctan(y/x)

解

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

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

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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