(\partial)/(\partial z)(1/(x^2+y^2+z^2))

Lösung

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

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

Schritte zur Lösung

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

Behandle

x, y

als Konstanten

Wende Exponentenregel an:

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

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

t a y l or e^{x^{2}}

in v erse l a pl a ce \frac{s - 1}{2 s ^{2} + s + 3}

\frac{d}{d x} (5 x e^{x^{2}})

\frac{d}{d t} (2 t e^{2 t})

\int x^{3} 121 - x^{2} d x