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

解

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

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

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{1}{- x + 1 ( x + 1 ) ^{\frac{3}{2}}}

= \frac{1}{( \frac{1 - x}{1 + x} ) ^{2} + 1} (- \frac{1}{- x + 1 ( x + 1 ) ^{\frac{3}{2}}})

簡素化

\frac{1}{( \frac{1 - x}{1 + x} ) ^{2} + 1} (- \frac{1}{- x + 1 ( x + 1 ) ^{\frac{3}{2}}}) : - \frac{1}{2 - x + 1 ( x + 1 ) ^{\frac{1}{2}}}

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