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Beliebt Rechnen >

(\partial)/(\partial x)((5xyz)/(x^2+y^2+z^2))

Lösung

\frac{\partial}{\partial x} (\frac{5 x yz}{x ^{2} + y ^{2} + z ^{2}})

Lösung

\frac{5 yz ( - x ^{2} + y ^{2} + z ^{2} )}{( x ^{2} + y ^{2} + z ^{2} ) ^{2}}

Schritte zur Lösung

\frac{\partial}{\partial x} (\frac{5 x yz}{x ^{2} + y ^{2} + z ^{2}})

Behandle

y, z

als Konstanten

Entferne die Konstante:

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

= 5 yz \frac{\partial}{\partial x} (\frac{x}{x ^{2} + y ^{2} + z ^{2}})

Wende die Quotientenregel an:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= 5 yz \frac{\frac{\partial}{\partial x} ( x ) ( x ^{2} + y ^{2} + z ^{2} ) - \frac{\partial}{\partial x} ( x ^{2} + y ^{2} + z ^{2} ) x}{( x ^{2} + y ^{2} + z ^{2} ) ^{2}}

\frac{\partial}{\partial x} (x) = 1

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

= 5 yz \frac{1 \cdot ( x ^{2} + y ^{2} + z ^{2} ) - 2 xx}{( x ^{2} + y ^{2} + z ^{2} ) ^{2}}

Vereinfache

5 yz \frac{1 \cdot ( x ^{2} + y ^{2} + z ^{2} ) - 2 xx}{( x ^{2} + y ^{2} + z ^{2} ) ^{2}} : \frac{5 yz ( - x ^{2} + y ^{2} + z ^{2} )}{( x ^{2} + y ^{2} + z ^{2} ) ^{2}}

= \frac{5 yz ( - x ^{2} + y ^{2} + z ^{2} )}{( x ^{2} + y ^{2} + z ^{2} ) ^{2}}

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