(\partial)/(\partial y)((x+y)(x^2+2y^2))

Lösung

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

6 y^{2} + 4 x y + x^{2}

Schritte zur Lösung

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

Behandle

x

als Konstante

Wende die Produktregel an:

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

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

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

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

\frac{\partial}{\partial y} (x^{2} + 2 y^{2}) = 4 y

= 1 \cdot (x^{2} + 2 y^{2}) + 4 y (x + y)

Vereinfache

1 \cdot (x^{2} + 2 y^{2}) + 4 y (x + y) : 6 y^{2} + 4 x y + x^{2}

= 6 y^{2} + 4 x y + x^{2}

\frac{d y}{d t} + e^{t} y = - 24 e^{t}

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