derivative f(x)=x^a(1-x)^b

解

導関数

f (x) = x^{a} (1 - x)^{b}

a x^{a - 1} (1 - x)^{b} - b x^{a} (1 - x)^{b - 1}

解答ステップ

\frac{d}{d x} (x^{a} (1 - x)^{b})

積の法則を適用:

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

f = x^{a}, g = (1 - x)^{b}

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

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

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

= a x^{a - 1} (1 - x)^{b} + (- b (1 - x)^{b - 1}) x^{a}

簡素化

= a x^{a - 1} (1 - x)^{b} - b x^{a} (1 - x)^{b - 1}