derivative arctan(1/x)

解

導関数

arctan (\frac{1}{x})

- \frac{1}{1 + x ^{2}}

解答ステップ

\frac{d}{d x} (arctan (\frac{1}{x}))

連鎖法則を適用:

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

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

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

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

簡素化

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

= - \frac{1}{1 + x ^{2}}