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Beliebt Rechnen >

(\partial)/(\partial y)(x/(1+x^2+y^2))

Lösung

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

Lösung

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

Schritte zur Lösung

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

Behandle

x

als Konstante

Entferne die Konstante:

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

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

Wende Exponentenregel an:

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

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

Beliebte Beispiele

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