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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{2 x y ^{3}}{( x ^{2} + y ^{2} ) ^{2}}

Schritte zur Lösung

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

Behandle

y

als Konstante

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

Vereinfache

y \frac{2 x ( x ^{2} + y ^{2} ) - 2 x x ^{2}}{( x ^{2} + y ^{2} ) ^{2}} : \frac{2 x y ^{3}}{( x ^{2} + y ^{2} ) ^{2}}

= \frac{2 x y ^{3}}{( x ^{2} + y ^{2} ) ^{2}}

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