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

解答

\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}}

求解步骤

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

使用微分除法定则:

(\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}}

化简

\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}}

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

t y^{^{'}} - 2 y = t^{3} e^{- t}

\int_{0}^{8} 1 + 36 x d x

t an g e n t f (x) = 2 x^{2} + 4 x, at x = - 2

\frac{d}{d x} (4 (3 x^{2} - 2))