derivative of (arcsin(x)/(arctan(x)))

解

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

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

解答ステップ

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

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

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

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

簡素化

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

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