derivative f(x)=xsqrt(x-x^2)

解

導関数

f (x) = x x - x^{2}

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

解答ステップ

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

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = x - x^{2}

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

\frac{d x}{d x} = 1

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

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

簡素化

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

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