derivative f(x)=-sqrt(1-x)x^2

解

導関数

f (x) = - 1 - x x^{2}

- \frac{- 5 x ^{2} + 4 x}{2 1 - x}

解答ステップ

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

定数を除去:

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

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

積の法則を適用:

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

f = 1 - x, g = x^{2}

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

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

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

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

簡素化

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

= - \frac{- 5 x ^{2} + 4 x}{2 1 - x}