derivative f(x)=(x^2)/(7+sqrt(x))

解

導関数

f (x) = \frac{x ^{2}}{7 + x}

\frac{4 x ( 7 + x ) - x ^{\frac{3}{2}}}{2 ( 7 + x ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{2}}{7 + 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} ) ( 7 + x ) - \frac{d}{d x} ( 7 + x ) x ^{2}}{( 7 + x ) ^{2}}

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

\frac{d}{d x} (7 + x) = \frac{1}{2 x}

= \frac{2 x ( 7 + x ) - \frac{1}{2 x} x ^{2}}{( 7 + x ) ^{2}}

簡素化

\frac{2 x ( 7 + x ) - \frac{1}{2 x} x ^{2}}{( 7 + x ) ^{2}} : \frac{4 x ( 7 + x ) - x ^{\frac{3}{2}}}{2 ( 7 + x ) ^{2}}

= \frac{4 x ( 7 + x ) - x ^{\frac{3}{2}}}{2 ( 7 + x ) ^{2}}