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

解

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

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

解答ステップ

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

\frac{d}{d x} (arcsin (2 x)) = \frac{2}{1 - 4 x ^{2}}

\frac{d}{d x} (arccos (2 x)) = - \frac{2}{1 - 4 x ^{2}}

= \frac{\frac{2}{1 - 4 x ^{2}} arccos ( 2 x ) - ( - \frac{2}{1 - 4 x ^{2}} ) arcsin ( 2 x )}{( arccos ( 2 x ) ) ^{2}}

簡素化

\frac{\frac{2}{1 - 4 x ^{2}} arccos ( 2 x ) - ( - \frac{2}{1 - 4 x ^{2}} ) arcsin ( 2 x )}{arccos ^{2} ( 2 x )} : \frac{2 arccos ( 2 x ) + 2 arcsin ( 2 x )}{arccos ^{2} ( 2 x ) 1 - 4 x ^{2}}

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