derivative of arccos(x/(sqrt(1-x^2)))

解

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

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

解答ステップ

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

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

簡素化

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

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