derivative of arcsin(x/(1+x^2))

解

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

\frac{1 - x ^{2}}{( x ^{2} + 1 ) x ^{4} + x ^{2} + 1}

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

\frac{1}{1 - ( \frac{x}{1 + x ^{2}} ) ^{2}} \cdot \frac{1 - x ^{2}}{( 1 + x ^{2} ) ^{2}} : \frac{1 - x ^{2}}{( x ^{2} + 1 ) x ^{4} + x ^{2} + 1}

= \frac{1 - x ^{2}}{( x ^{2} + 1 ) x ^{4} + x ^{2} + 1}