(\partial)/(\partial x)(9xe^{x^2+y^2})

Lösung

\frac{\partial}{\partial x} (9 x e^{x^{2} + y^{2}})

9 (e^{x^{2} + y^{2}} + 2 e^{x^{2} + y^{2}} x^{2})

Schritte zur Lösung

\frac{\partial}{\partial x} (9 x e^{x^{2} + y^{2}})

Behandle

y

als Konstante

Entferne die Konstante:

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

= 9 \frac{\partial}{\partial x} (x e^{x^{2} + y^{2}})

Wende die Produktregel an:

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

f = x, g = e^{x^{2} + y^{2}}

= 9 (\frac{\partial}{\partial x} (x) e^{x^{2} + y^{2}} + \frac{\partial}{\partial x} (e^{x^{2} + y^{2}}) x)

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

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

= 9 (1 \cdot e^{x^{2} + y^{2}} + e^{x^{2} + y^{2}} \cdot 2 xx)

Vereinfache

9 (1 \cdot e^{x^{2} + y^{2}} + e^{x^{2} + y^{2}} \cdot 2 xx) : 9 (e^{x^{2} + y^{2}} + 2 e^{x^{2} + y^{2}} x^{2})

= 9 (e^{x^{2} + y^{2}} + 2 e^{x^{2} + y^{2}} x^{2})

\frac{d}{d x} (\frac{1}{5} x^{5} - 2 x^{3} + 3 x)

x \to - 1 lim (6 x + 3)

x \to 1 lim (\frac{e ^{x}}{1 - x})

x \to 1 lim (\frac{1 ( x )}{1 - 3 x})

s \to 0 lim (\frac{1}{s})