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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Behandle

y

als Konstante

Entferne die Konstante:

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

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

Wende Exponentenregel an:

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

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

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