derivative of (xe^x-e^x/(x^2))

解答

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

\frac{e ^{x} ( x ^{2} - 2 ( x - 1 ) )}{x ^{3}}

求解步骤

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

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

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

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

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

化简

\frac{e ^{x} x x ^{2} - 2 x ( x e ^{x} - e ^{x} )}{( x ^{2} ) ^{2}} : \frac{e ^{x} ( x ^{2} - 2 ( x - 1 ) )}{x ^{3}}

= \frac{e ^{x} ( x ^{2} - 2 ( x - 1 ) )}{x ^{3}}

x \to 0 lim (3 x + 1)

t an g e n t \frac{4}{x - 7}

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

l a pl a ce t r an s f or m f (t) = - 4 t^{2} + 16 t + 9

\frac{d}{d x} ((3 x^{2} - 4 x) e^{x})