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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Behandle

y, z

als Konstanten

Entferne die Konstante:

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

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

Wende Exponentenregel an:

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

= yz \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

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

Vereinfache

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

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

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