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

解

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

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

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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