derivative (x^2)/(5(4-x))

解答

导数

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

\frac{8 x - x ^{2}}{5 ( 4 - x ) ^{2}}

求解步骤

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

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{1}{5} \frac{d}{d x} (\frac{x ^{2}}{4 - x})

使用微分除法定则:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{1}{5} \cdot \frac{\frac{d}{d x} ( x ^{2} ) ( 4 - x ) - \frac{d}{d x} ( 4 - x ) x ^{2}}{( 4 - x ) ^{2}}

\frac{d}{d x} (x^{2}) = 2 x

\frac{d}{d x} (4 - x) = - 1

= \frac{1}{5} \cdot \frac{2 x ( 4 - x ) - ( - 1 ) x ^{2}}{( 4 - x ) ^{2}}

化简

\frac{1}{5} \cdot \frac{2 x ( 4 - x ) - ( - 1 ) x ^{2}}{( 4 - x ) ^{2}} : \frac{8 x - x ^{2}}{5 ( 4 - x ) ^{2}}

= \frac{8 x - x ^{2}}{5 ( 4 - x ) ^{2}}