derivative of (3x^{2x})

解答

\frac{d}{d x} ((3 x)^{2 x})

2 ln (3) \cdot 9^{x} x^{2 x} + 2 x^{2 x} (ln (x) + 1) \cdot 9^{x}

求解步骤

\frac{d}{d x} ((3 x)^{2 x})

化简

(3 x)^{2 x} : 9^{x} x^{2 x}

= \frac{d}{d x} (9^{x} x^{2 x})

使用微分乘法定则:

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

f = 9^{x}, g = x^{2 x}

= \frac{d}{d x} (9^{x}) x^{2 x} + \frac{d}{d x} (x^{2 x}) 9^{x}

\frac{d}{d x} (9^{x}) = 2 ln (3) \cdot 9^{x}

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

= 2 ln (3) \cdot 9^{x} x^{2 x} + 2 x^{2 x} (ln (x) + 1) \cdot 9^{x}

n = 0 \sum \infty \frac{n}{e ^{n^{2}}}

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

\int ln (1 + x^{2}) d x

\frac{\partial}{\partial x} ((2 x + 8) ln (x y))

f^{^{'}} (x) = 3 x + 2