(\partial)/(\partial x)(xe^{-2x^2-2y^2})

Lösung

\frac{\partial}{\partial x} (x e^{- 2 x^{2} - 2 y^{2}})

e^{- 2 x^{2} - 2 y^{2}} - 4 e^{- 2 x^{2} - 2 y^{2}} x^{2}

Schritte zur Lösung

\frac{\partial}{\partial x} (x e^{- 2 x^{2} - 2 y^{2}})

Behandle

y

als Konstante

Wende die Produktregel an:

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

f = x, g = e^{- 2 x^{2} - 2 y^{2}}

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

\frac{\partial}{\partial x} (x) = 1

\frac{\partial}{\partial x} (e^{- 2 x^{2} - 2 y^{2}}) = - 4 e^{- 2 x^{2} - 2 y^{2}} x

= 1 \cdot e^{- 2 x^{2} - 2 y^{2}} + (- 4 e^{- 2 x^{2} - 2 y^{2}} x) x

Vereinfache

1 \cdot e^{- 2 x^{2} - 2 y^{2}} + (- 4 e^{- 2 x^{2} - 2 y^{2}} x) x : e^{- 2 x^{2} - 2 y^{2}} - 4 e^{- 2 x^{2} - 2 y^{2}} x^{2}

= e^{- 2 x^{2} - 2 y^{2}} - 4 e^{- 2 x^{2} - 2 y^{2}} x^{2}