derivative of (1+ln(x)/(1-ln(x)))

解答

\frac{d}{d x} (\frac{1 + ln ( x )}{1 - ln ( x )})

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

求解步骤

\frac{d}{d x} (\frac{1 + ln ( x )}{1 - ln ( 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} ( 1 + ln ( x ) ) ( 1 - ln ( x ) ) - \frac{d}{d x} ( 1 - ln ( x ) ) ( 1 + ln ( x ) )}{( 1 - ln ( x ) ) ^{2}}

\frac{d}{d x} (1 + ln (x)) = \frac{1}{x}

\frac{d}{d x} (1 - ln (x)) = - \frac{1}{x}

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

化简

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

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

\int \frac{4}{4 - x ^{2}} d x

t an g e n t - \frac{6 x}{x ^{2} + 1}, at x = 1

d er i v a t i v e f (x) = 9 x^{2}

\int \frac{x}{1 - 5 x ^{2}} d x

\frac{d}{d x} (\frac{1}{x ^{13}})