derivative of (ax^2+bx+ce^{(-x)/6})

Lösung

\frac{d}{d x} ((a x^{2} + b x + c) e^{\frac{- x}{6}})

\frac{- a e ^{- \frac{x}{6}} x ^{2} - b e ^{- \frac{x}{6}} x - c e ^{- \frac{x}{6}}}{6} + 2 a e^{- \frac{x}{6}} x + b e^{- \frac{x}{6}}

Schritte zur Lösung

\frac{d}{d x} ((a x^{2} + b x + c) e^{\frac{- x}{6}})

Vereinfache

(a x^{2} + b x + c) e^{\frac{- x}{6}} : e^{- \frac{x}{6}} (a x^{2} + b x + c)

= \frac{d}{d x} (e^{- \frac{x}{6}} (a x^{2} + b x + c))

Wende die Produktregel an:

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

f = e^{- \frac{x}{6}}, g = a x^{2} + b x + c

= \frac{d}{d x} (e^{- \frac{x}{6}}) (a x^{2} + b x + c) + \frac{d}{d x} (a x^{2} + b x + c) e^{- \frac{x}{6}}

\frac{d}{d x} (e^{- \frac{x}{6}}) = - \frac{1}{6} e^{- \frac{x}{6}}

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

= (- \frac{1}{6} e^{- \frac{x}{6}}) (a x^{2} + b x + c) + (2 a x + b) e^{- \frac{x}{6}}

Vereinfache

(- \frac{1}{6} e^{- \frac{x}{6}}) (a x^{2} + b x + c) + (2 a x + b) e^{- \frac{x}{6}} : \frac{- a e ^{- \frac{x}{6}} x ^{2} - b e ^{- \frac{x}{6}} x - c e ^{- \frac{x}{6}}}{6} + 2 a e^{- \frac{x}{6}} x + b e^{- \frac{x}{6}}

= \frac{- a e ^{- \frac{x}{6}} x ^{2} - b e ^{- \frac{x}{6}} x - c e ^{- \frac{x}{6}}}{6} + 2 a e^{- \frac{x}{6}} x + b e^{- \frac{x}{6}}

x \to 7 lim (\frac{5 x}{x + 2})

t \to - 2 lim (t^{2} - 6 t - 16)

y^{^{''}} - 2 y^{^{'}} + y = e^{t} arctan (t)

\int \frac{s}{2 s + 3} d s

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