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

Lösung

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

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

Schritte zur Lösung

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

Behandle

y

als Konstante

Entferne die Konstante:

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

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

Wende die Produktregel an:

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

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

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

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

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

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

Vereinfache

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

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

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