derivative of (arcsin(x)/x)

解

\frac{d}{d x} (\frac{arcsin ( x )}{x})

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

解答ステップ

\frac{d}{d x} (\frac{arcsin ( x )}{x})

商の法則を適用:

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

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

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

\frac{d x}{d x} = 1

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

簡素化

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

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