derivative of (ln(x^2)/(e^{2x)})

解答

\frac{d}{d x} (\frac{ln ( x ^{2} )}{e ^{2 x}})

\frac{2 ( 1 - 2 x ln ( x ) )}{e ^{2 x} x}

求解步骤

\frac{d}{d x} (\frac{ln ( x ^{2} )}{e ^{2 x}})

使用微分除法定则:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( ln ( x ^{2} ) ) e ^{2 x} - \frac{d}{d x} ( e ^{2 x} ) ln ( x ^{2} )}{( e ^{2 x} ) ^{2}}

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

\frac{d}{d x} (e^{2 x}) = e^{2 x} \cdot 2

= \frac{\frac{2}{x} e ^{2 x} - e ^{2 x} \cdot 2 ln ( x ^{2} )}{( e ^{2 x} ) ^{2}}

化简

\frac{\frac{2}{x} e ^{2 x} - e ^{2 x} \cdot 2 ln ( x ^{2} )}{( e ^{2 x} ) ^{2}} : \frac{2 ( 1 - 2 x ln ( x ) )}{e ^{2 x} x}

= \frac{2 ( 1 - 2 x ln ( x ) )}{e ^{2 x} x}

n = 0 \sum \infty \frac{1}{n ^{2} + 3 n + 2}

x \to 0 lim (\frac{e ^{x} + e ^{- x}}{e ^{x} - e ^{- x}})

\frac{d}{d x} (1 - x)

s l o p e (0, - 4), (1, 2)

(1 - 5 y + x) \frac{d y}{d x} + y = 5 x - 1