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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Vereinfache

\frac{- y}{x ^{2} + y ^{2}} : - \frac{y}{x ^{2} + y ^{2}}

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

Behandle

y

als Konstante

Entferne die Konstante:

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

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

Wende Exponentenregel an:

\frac{1}{a} = a^{- 1}

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

Wende die Kettenregel an:

- \frac{1}{( x ^{2} + y ^{2} ) ^{2}} \frac{\partial}{\partial x} (x^{2} + y^{2})

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

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

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

Vereinfache

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

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

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