derivative of (2x+3x^2/(x^2e^x))

Lösung

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

- \frac{3 x ^{2} + 2 x + 2}{e ^{x} x ^{2}}

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

\frac{d}{d x} (2 x + 3 x^{2}) = 2 + 6 x

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

= \frac{( 2 + 6 x ) x ^{2} e ^{x} - ( 2 x e ^{x} + e ^{x} x ^{2} ) ( 2 x + 3 x ^{2} )}{( x ^{2} e ^{x} ) ^{2}}

Vereinfache

\frac{( 2 + 6 x ) x ^{2} e ^{x} - ( 2 x e ^{x} + e ^{x} x ^{2} ) ( 2 x + 3 x ^{2} )}{( x ^{2} e ^{x} ) ^{2}} : - \frac{3 x ^{2} + 2 x + 2}{e ^{x} x ^{2}}

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

l a pl a ce t r an s f or m t e^{- 3 t} cos (3 t)

\frac{d}{d x} (\frac{80 - x}{4})

\int x sin^{2} (2 x) d x

d er i v a t i v e 4 t^{3}

d er i v a t i v e [x + (x + sin^{2} (x))^{3}]^{6}