derivative 0.5x^34^x

解答

导数

0.5 x^{3} 4^{x}

0.5 (3 \cdot 4^{x} x^{2} + 0.69314 \dots \cdot 2^{2 x + 1} x^{3})

求解步骤

\frac{d}{d x} (0.5 x^{3} \cdot 4^{x})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 0.5 \frac{d}{d x} (x^{3} \cdot 4^{x})

使用微分乘法定则:

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

f = x^{3}, g = 4^{x}

= 0.5 (\frac{d}{d x} (x^{3}) 4^{x} + \frac{d}{d x} (4^{x}) x^{3})

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

\frac{d}{d x} (4^{x}) = ln (2) \cdot 2^{2 x + 1}

= 0.5 (3 x^{2} \cdot 4^{x} + ln (2) \cdot 2^{2 x + 1} x^{3})

整理为小数形式

= 0.5 (3 \cdot 4^{x} x^{2} + 0.69314 \dots \cdot 2^{2 x + 1} x^{3})

\int_{0}^{\frac{1}{13}} x ln (13 x) d x

y^{^{''}} - 12 y^{^{'}} + 45 y = 0

\frac{d}{d x} ((x - sin (x)) (x + cos (x)))

2 x \frac{d y}{d x} + y^{3} e^{- 2 x} = 2 x y

s im pl i f y cos (x y)