derivative of (x(x+1)/(2x-1))

解答

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

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

求解步骤

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

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

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

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

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

乘开

(2 x + 1) (2 x - 1) - 2 x (x + 1) : 2 x^{2} - 2 x - 1

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

\int \frac{1}{x ( x + 1 )} d x

d er i v a t i v e y = 2 - x^{2}

\int_{- 2 π}^{π} π sin^{2} (x) d x

\frac{d y}{d x} = \frac{x ^{2} - x + 2}{x} + y

\int 5 sin (5 x - 3) d x