derivative F(x)=(x^2)/(4+sqrt(x))

解

derivative

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

解答ステップ

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

商の法則を適用:

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

= \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) = \frac{1}{2 x}

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

簡素化

\frac{2 x ( 4 + x ) - \frac{1}{2 x} x ^{2}}{( 4 + x ) ^{2}} : \frac{4 x ( 4 + x ) - x ^{\frac{3}{2}}}{2 ( 4 + x ) ^{2}}

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