derivative f(x)=(x^5-5x^2)/(2x^5+5x)

解答

导数

f (x) = \frac{x ^{5} - 5 x ^{2}}{2 x ^{5} + 5 x}

\frac{5 ( 6 x ^{4} + 4 x ^{3} - 5 )}{( 2 x ^{4} + 5 ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{x ^{5} - 5 x ^{2}}{2 x ^{5} + 5 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} ( x ^{5} - 5 x ^{2} ) ( 2 x ^{5} + 5 x ) - \frac{d}{d x} ( 2 x ^{5} + 5 x ) ( x ^{5} - 5 x ^{2} )}{( 2 x ^{5} + 5 x ) ^{2}}

\frac{d}{d x} (x^{5} - 5 x^{2}) = 5 x^{4} - 10 x

\frac{d}{d x} (2 x^{5} + 5 x) = 10 x^{4} + 5

= \frac{( 5 x ^{4} - 10 x ) ( 2 x ^{5} + 5 x ) - ( 10 x ^{4} + 5 ) ( x ^{5} - 5 x ^{2} )}{( 2 x ^{5} + 5 x ) ^{2}}

化简

\frac{( 5 x ^{4} - 10 x ) ( 2 x ^{5} + 5 x ) - ( 10 x ^{4} + 5 ) ( x ^{5} - 5 x ^{2} )}{( 2 x ^{5} + 5 x ) ^{2}} : \frac{5 ( 6 x ^{4} + 4 x ^{3} - 5 )}{( 2 x ^{4} + 5 ) ^{2}}

= \frac{5 ( 6 x ^{4} + 4 x ^{3} - 5 )}{( 2 x ^{4} + 5 ) ^{2}}

\frac{\partial}{\partial x} (3 x z^{3} e^{y^{2}})

\int (ln (2 x))^{2} d x

x \to - 3 lim (\frac{2}{x + 2})

\int x (x + 3) d x

d er i v a t i v e \frac{x}{x - 6}