(\partial)/(\partial x)((1+xy)e^{xy})

Lösung

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

2 e^{x y} y + e^{x y} x y^{2}

Schritte zur Lösung

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

Behandle

y

als Konstante

Wende die Produktregel an:

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

f = 1 + x y, g = e^{x y}

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

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

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

= y e^{x y} + e^{x y} y (1 + x y)

Vereinfache

y e^{x y} + e^{x y} y (1 + x y) : 2 e^{x y} y + e^{x y} x y^{2}

= 2 e^{x y} y + e^{x y} x y^{2}

y^{^{'}} = 1 - x + y - x y

n = 1 \sum \infty \frac{4 ^{n}}{n + 1}

d er i v a t i v e f (x) = 4 x^{4} - 3 x^{\frac{5}{2}} + 2

\frac{\partial}{\partial x} (3 e^{y} + x + x^{3} + 6)

y^{^{'}} = 2 x y^{2}