derivative f(x)=(8x^3-3x+2)/(3x-x^4)

解答

导数

f (x) = \frac{8 x ^{3} - 3 x + 2}{3 x - x ^{4}}

\frac{8 x ^{6} - 9 x ^{4} + 56 x ^{3} - 6}{( 3 x - x ^{4} ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{8 x ^{3} - 3 x + 2}{3 x - x ^{4}})

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} ( 8 x ^{3} - 3 x + 2 ) ( 3 x - x ^{4} ) - \frac{d}{d x} ( 3 x - x ^{4} ) ( 8 x ^{3} - 3 x + 2 )}{( 3 x - x ^{4} ) ^{2}}

\frac{d}{d x} (8 x^{3} - 3 x + 2) = 24 x^{2} - 3

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

= \frac{( 24 x ^{2} - 3 ) ( 3 x - x ^{4} ) - ( 3 - 4 x ^{3} ) ( 8 x ^{3} - 3 x + 2 )}{( 3 x - x ^{4} ) ^{2}}

乘开

(24 x^{2} - 3) (3 x - x^{4}) - (3 - 4 x^{3}) (8 x^{3} - 3 x + 2) : 8 x^{6} - 9 x^{4} + 56 x^{3} - 6

= \frac{8 x ^{6} - 9 x ^{4} + 56 x ^{3} - 6}{( 3 x - x ^{4} ) ^{2}}

(cos (x^{2}) \cdot 2 x)^{^{'}}

\frac{\partial}{\partial x} (ln (x^{2} + y^{2} + x y))

\frac{\partial}{\partial y} (2 k y - 5 x)

\int y^{- \frac{1}{3}} d y

\int \frac{5 x - 12}{x ^{3} - 6 x ^{2} + 8 x} d x