derivative of (arccos(x)/(x^2))

解

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

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

解答ステップ

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

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

簡素化

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

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