d/(d{y)}(({z})/(({x)^2+{y}^2+{z}^2)^{3/2}})

Lösung

\frac{d}{d y} (\frac{z}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}})

- \frac{3 z y}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}}

Schritte zur Lösung

\frac{d}{d y} (\frac{z}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}})

Entferne die Konstante:

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

= z \frac{d}{d y} (\frac{1}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}})

Wende Exponentenregel an:

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

= z \frac{d}{d y} ((x^{2} + y^{2} + z^{2})^{- \frac{3}{2}})

Wende die Kettenregel an:

- \frac{3}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}} \frac{d}{d y} (x^{2} + y^{2} + z^{2})

= - \frac{3}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}} \frac{d}{d y} (x^{2} + y^{2} + z^{2})

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

= z (- \frac{3}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}} \cdot 2 y)

Vereinfache

z (- \frac{3}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}} \cdot 2 y) : - \frac{3 z y}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}}

= - \frac{3 z y}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{5}{2}}}

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