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

Lösung

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

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

Schritte zur Lösung

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

Behandle

x, y

als Konstanten

Wende die Quotientenregel an:

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

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

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

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

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

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

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