(\partial)/(\partial x)((x^2+y^2+z^2)^{-0.5})

Lösung

\frac{\partial}{\partial x} ((x^{2} + y^{2} + z^{2})^{- 0.5})

- \frac{x}{( x ^{2} + y ^{2} + z ^{2} ) ^{1.5}}

Schritte zur Lösung

\frac{\partial}{\partial x} ((x^{2} + y^{2} + z^{2})^{- 0.5})

Behandle

y, z

als Konstanten

Wende die Kettenregel an:

- \frac{0.5}{( x ^{2} + y ^{2} + z ^{2} ) ^{1.5}} \frac{\partial}{\partial x} (x^{2} + y^{2} + z^{2})

= - \frac{0.5}{( x ^{2} + y ^{2} + z ^{2} ) ^{1.5}} \frac{\partial}{\partial x} (x^{2} + y^{2} + z^{2})

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

= - \frac{0.5}{( x ^{2} + y ^{2} + z ^{2} ) ^{1.5}} \cdot 2 x

Vereinfache

- \frac{0.5}{( x ^{2} + y ^{2} + z ^{2} ) ^{1.5}} \cdot 2 x : - \frac{x}{( x ^{2} + y ^{2} + z ^{2} ) ^{1.5}}

= - \frac{x}{( x ^{2} + y ^{2} + z ^{2} ) ^{1.5}}

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