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

Lösung

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

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

Schritte zur Lösung

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

Behandle

y

als Konstante

Wende die Produktregel an:

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

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

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

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

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

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

Vereinfache

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

x \to 3 + lim (4)

\int \frac{3}{5 x ^{3}} d x

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

x^{2} \frac{d y}{d x} + y = 0

\frac{d}{d x} (sin^{2} (6 x))