derivative of (ax^2+bx+ce^{(-x)/2})

Lösung

\frac{d}{d x} ((a x^{2} + b x + c) e^{\frac{- x}{2}})

\frac{- a e ^{- \frac{x}{2}} x ^{2} - b e ^{- \frac{x}{2}} x - c e ^{- \frac{x}{2}}}{2} + 2 a e^{- \frac{x}{2}} x + b e^{- \frac{x}{2}}

Schritte zur Lösung

\frac{d}{d x} ((a x^{2} + b x + c) e^{\frac{- x}{2}})

Vereinfache

(a x^{2} + b x + c) e^{\frac{- x}{2}} : e^{- \frac{x}{2}} (a x^{2} + b x + c)

= \frac{d}{d x} (e^{- \frac{x}{2}} (a x^{2} + b x + c))

Wende die Produktregel an:

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

f = e^{- \frac{x}{2}}, g = a x^{2} + b x + c

= \frac{d}{d x} (e^{- \frac{x}{2}}) (a x^{2} + b x + c) + \frac{d}{d x} (a x^{2} + b x + c) e^{- \frac{x}{2}}

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

\frac{d}{d x} (a x^{2} + b x + c) = 2 a x + b

= (- \frac{1}{2} e^{- \frac{x}{2}}) (a x^{2} + b x + c) + (2 a x + b) e^{- \frac{x}{2}}

Vereinfache

(- \frac{1}{2} e^{- \frac{x}{2}}) (a x^{2} + b x + c) + (2 a x + b) e^{- \frac{x}{2}} : \frac{- a e ^{- \frac{x}{2}} x ^{2} - b e ^{- \frac{x}{2}} x - c e ^{- \frac{x}{2}}}{2} + 2 a e^{- \frac{x}{2}} x + b e^{- \frac{x}{2}}

= \frac{- a e ^{- \frac{x}{2}} x ^{2} - b e ^{- \frac{x}{2}} x - c e ^{- \frac{x}{2}}}{2} + 2 a e^{- \frac{x}{2}} x + b e^{- \frac{x}{2}}

d er i v a t i v e \frac{1}{3 s ^{2}} - \frac{5}{4 s ^{3}}

\frac{d}{d x} (arcsin (2 x) + arccos (2 x))

y^{^{''}} - y^{^{'}} - 6 y = 2 e^{2 t} - t^{2} - 2 t + 3

x \to 6 lim (\frac{x ^{2} + 2 x - 48}{3 x ^{2} - 20 x + 12})

s im pl i f y \frac{- ln ( 1 + 6 x )}{x}