derivative of (ax^2+bx+ce^{-x/8})

Lösung

\frac{d}{d x} ((a x^{2} + b x + c) e^{- \frac{x}{8}})

2 a e^{- \frac{x}{8}} x + b e^{- \frac{x}{8}} + \frac{- a e ^{- \frac{x}{8}} x ^{2} - b e ^{- \frac{x}{8}} x - c e ^{- \frac{x}{8}}}{8}

Schritte zur Lösung

\frac{d}{d x} ((a x^{2} + b x + c) e^{- \frac{x}{8}})

Wende die Produktregel an:

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

f = a x^{2} + b x + c, g = e^{- \frac{x}{8}}

= \frac{d}{d x} (a x^{2} + b x + c) e^{- \frac{x}{8}} + \frac{d}{d x} (e^{- \frac{x}{8}}) (a x^{2} + b x + c)

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

\frac{d}{d x} (e^{- \frac{x}{8}}) = - \frac{1}{8} e^{- \frac{x}{8}}

= (2 a x + b) e^{- \frac{x}{8}} + (- \frac{1}{8} e^{- \frac{x}{8}}) (a x^{2} + b x + c)

Vereinfache

(2 a x + b) e^{- \frac{x}{8}} + (- \frac{1}{8} e^{- \frac{x}{8}}) (a x^{2} + b x + c) : 2 a e^{- \frac{x}{8}} x + b e^{- \frac{x}{8}} + \frac{- a e ^{- \frac{x}{8}} x ^{2} - b e ^{- \frac{x}{8}} x - c e ^{- \frac{x}{8}}}{8}

= 2 a e^{- \frac{x}{8}} x + b e^{- \frac{x}{8}} + \frac{- a e ^{- \frac{x}{8}} x ^{2} - b e ^{- \frac{x}{8}} x - c e ^{- \frac{x}{8}}}{8}

x \to - 1 lim (- x^{2} + 4 x)

\int 1 - 9 x^{2} d x

\int_{0}^{\frac{π}{2}} x cos (2 x) d x

\frac{\partial}{\partial x} (x^{y} - e^{x y})

x \to + 0 lim ((x + 1)^{2} - x^{2})