derivative f(x)=(x^2)/(x+1)

解

導関数

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

\frac{x ^{2} + 2 x}{( x + 1 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{2}}{x + 1})

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

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

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

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

簡素化

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

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