(\partial)/(\partial x)(xye^{x/y})

Lösung

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

e^{\frac{x}{y}} y + e^{\frac{x}{y}} x

Schritte zur Lösung

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

Behandle

y

als Konstante

Entferne die Konstante:

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

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

Wende die Produktregel an:

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

f = x, g = e^{\frac{x}{y}}

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

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

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

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

Vereinfache

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

= e^{\frac{x}{y}} y + e^{\frac{x}{y}} x