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

Lösung

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

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

Schritte zur Lösung

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

Behandle

y

als Konstante

Entferne die Konstante:

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

= 2 \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}}

= 2 (\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

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

Vereinfache

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

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

x \to 0 lim (x (1 - \frac{1}{x}))

\frac{d ^{2}}{d x ^{2}} f (x) = 4 x^{5} - \frac{3}{x ^{2}}

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

\frac{d}{d x} (\frac{x ^{2}}{cos ( x )})

\int \frac{1}{2} e^{2 x} cos (x) d x