derivative of 1/(arctan(x))

解

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

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

解答ステップ

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

指数の規則を適用する:

\frac{1}{a} = a^{- 1}

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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