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

Lösung

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

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

Schritte zur Lösung

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

Behandle

y, z

als Konstanten

Wende die Produktregel an:

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

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

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

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

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

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

Vereinfache

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

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

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