d/(dz)((az^2)/(a^2+z^2))

Lösung

\frac{d}{d z} (\frac{a z ^{2}}{a ^{2} + z ^{2}})

\frac{2 a ^{3} z}{( a ^{2} + z ^{2} ) ^{2}}

Schritte zur Lösung

\frac{d}{d z} (\frac{a z ^{2}}{a ^{2} + z ^{2}})

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

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

= a \frac{\frac{d}{d z} ( z ^{2} ) ( a ^{2} + z ^{2} ) - \frac{d}{d z} ( a ^{2} + z ^{2} ) z ^{2}}{( a ^{2} + z ^{2} ) ^{2}}

\frac{d}{d z} (z^{2}) = 2 z

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

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

Vereinfache

a \frac{2 z ( a ^{2} + z ^{2} ) - 2 z z ^{2}}{( a ^{2} + z ^{2} ) ^{2}} : \frac{2 a ^{3} z}{( a ^{2} + z ^{2} ) ^{2}}

= \frac{2 a ^{3} z}{( a ^{2} + z ^{2} ) ^{2}}

\int x^{3} x^{2} - 7 d x

\int \frac{1}{x ^{3} 9 x ^{2} - 1} d x

2 y^{^{''}} + 3 y^{^{'}} - 4 y = 5 e^{3 x}

n = 1 \sum \infty \frac{( 3 x - 2 ) ^{n}}{n}

n = 1 \sum \infty \frac{7 n}{5 n + 1}