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

解答

导数

f (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}

t an g e n t f (x) = 9 - 4 x + 6 x^{2}, at x = 8

\frac{\partial}{\partial y} (y^{2} e^{x y^{2}})

\int z sin (x) d y

t an g e n t \frac{1}{3} x^{3} + \frac{7}{2} x^{2} - 3 x + 5

f (t) = tan (t)