derivative of-xe^{-(x^2/2})

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die Produktregel an:

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

f = x, g = e^{- \frac{x ^{2}}{2}}

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

\frac{d x}{d x} = 1

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

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

Vereinfache

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

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

f^{^{'}} (x) = 2 sec (x) + 3 tan (x) - tan (\frac{π}{3})

y^{^{''}} + (\frac{1}{x}) y^{^{'}} = 0

y^{^{'}} = \frac{- ( x y + 1 )}{x y + 1}

y^{^{'}} = (x - 3) (y + 1)^{\frac{2}{3}}

\int_{0}^{\frac{π}{2}} - 8 sin^{2} (3 x) d x