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

解

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

\frac{2 ( 1 - x ^{2} )}{x ^{4} + 6 x ^{2} + 1}

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

\frac{1}{( \frac{2 x}{1 + x ^{2}} ) ^{2} + 1} \cdot \frac{2 ( 1 - x ^{2} )}{( 1 + x ^{2} ) ^{2}} : \frac{2 ( 1 - x ^{2} )}{x ^{4} + 6 x ^{2} + 1}

= \frac{2 ( 1 - x ^{2} )}{x ^{4} + 6 x ^{2} + 1}