derivative of (x-2/(5x^2e^x))

Lösung

\frac{d}{d x} (\frac{x - 2}{5 x ^{2} e ^{x}})

\frac{e ^{x} x ^{2} - ( 2 e ^{x} x + e ^{x} x ^{2} ) ( x - 2 )}{5 e ^{2 x} x ^{4}}

Schritte zur Lösung

\frac{d}{d x} (\frac{x - 2}{5 x ^{2} e ^{x}})

Entferne die Konstante:

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

= \frac{1}{5} \frac{d}{d x} (\frac{x - 2}{x ^{2} e ^{x}})

Wende die Quotientenregel an:

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

= \frac{1}{5} \cdot \frac{\frac{d}{d x} ( x - 2 ) x ^{2} e ^{x} - \frac{d}{d x} ( x ^{2} e ^{x} ) ( x - 2 )}{( x ^{2} e ^{x} ) ^{2}}

\frac{d}{d x} (x - 2) = 1

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

= \frac{1}{5} \cdot \frac{1 \cdot x ^{2} e ^{x} - ( 2 x e ^{x} + e ^{x} x ^{2} ) ( x - 2 )}{( x ^{2} e ^{x} ) ^{2}}

Vereinfache

\frac{1}{5} \cdot \frac{1 \cdot x ^{2} e ^{x} - ( 2 x e ^{x} + e ^{x} x ^{2} ) ( x - 2 )}{( x ^{2} e ^{x} ) ^{2}} : \frac{e ^{x} x ^{2} - ( 2 e ^{x} x + e ^{x} x ^{2} ) ( x - 2 )}{5 e ^{2 x} x ^{4}}

= \frac{e ^{x} x ^{2} - ( 2 e ^{x} x + e ^{x} x ^{2} ) ( x - 2 )}{5 e ^{2 x} x ^{4}}

n = 1 \sum \infty \frac{1}{n ^{2} + 4 n + 3}

\int_{- 2}^{3} \frac{32}{x ^{4}} d x

l a pl a ce t r an s f or m e^{2 t} cos (t)

l a pl a ce t r an s f or m - 2 sin (3 t)

x \to \frac{π}{2} lim (\frac{cos ( x )}{cot ( x )})