derivative of a^x*x^a

解答

\frac{d}{d x} (a^{x} \cdot x^{a})

a^{x} x^{a} ln (a) + a^{1 + x} x^{a - 1}

求解步骤

\frac{d}{d x} (a^{x} x^{a})

使用微分乘法定则:

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

f = a^{x}, g = x^{a}

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

\frac{d}{d x} (a^{x}) = a^{x} ln (a)

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

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

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

a a^{x} = a^{1 + x}

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

x + 2 x y^{2} + e^{x^{2}} y \frac{d y}{d x} = 0

\int 2 (y - 1) d y

\int x^{2} cos (n x) d x

\int_{- 3}^{0} x^{3} d x

d er i v a t i v e 3 t^{2} - 27