derivative of arctan((1+x/(1-x)))

解

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

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

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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