derivative of x*e^{xy}

Lösung

\frac{d}{d x} (x \cdot e^{x y})

e^{x y} + e^{x y} x (y + x \frac{d y}{d x})

Schritte zur Lösung

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

Wende die Produktregel an:

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

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

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

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

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

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

Vereinfache

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