derivative of arctan(d/x)

解

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

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

解答ステップ

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

d

を定数として扱う

連鎖法則を適用:

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

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

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

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

簡素化

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

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