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

Lösung

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

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

Schritte zur Lösung

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

Behandle

x, z

als Konstanten

Entferne die Konstante:

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

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

Wende Exponentenregel an:

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

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

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