(\partial)/(\partial x)(xye^{-8(x^2+y^2)})

Lösung

\frac{\partial}{\partial x} (x y e^{- 8 (x^{2} + y^{2})})

y (e^{- 8 (x^{2} + y^{2})} - 16 e^{- 8 (x^{2} + y^{2})} x^{2})

Schritte zur Lösung

\frac{\partial}{\partial x} (x y e^{- 8 (x^{2} + y^{2})})

Behandle

y

als Konstante

Entferne die Konstante:

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

= y \frac{\partial}{\partial x} (x e^{- 8 (x^{2} + y^{2})})

Wende die Produktregel an:

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

f = x, g = e^{- 8 (x^{2} + y^{2})}

= y (\frac{\partial}{\partial x} (x) e^{- 8 (x^{2} + y^{2})} + \frac{\partial}{\partial x} (e^{- 8 (x^{2} + y^{2})}) x)

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

\frac{\partial}{\partial x} (e^{- 8 (x^{2} + y^{2})}) = - 16 e^{- 8 (x^{2} + y^{2})} x

= y (1 \cdot e^{- 8 (x^{2} + y^{2})} + (- 16 e^{- 8 (x^{2} + y^{2})} x) x)

Vereinfache

y (1 \cdot e^{- 8 (x^{2} + y^{2})} + (- 16 e^{- 8 (x^{2} + y^{2})} x) x) : y (e^{- 8 (x^{2} + y^{2})} - 16 e^{- 8 (x^{2} + y^{2})} x^{2})

= y (e^{- 8 (x^{2} + y^{2})} - 16 e^{- 8 (x^{2} + y^{2})} x^{2})

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