derivative of x^a(1-x^b)

解答

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

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

求解步骤

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

使用微分乘法定则:

(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}

化简

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

x \to - 8 lim (\frac{- 5 x}{x + 8})

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