erweitern (derivative of z^2)/((z+i)^2)

Lösung

erweitern

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

\frac{- \frac{d}{d x} ( z ^{2} ) + z ^{2} \frac{d}{d x} ( z ^{2} )}{1 + z ^{4} + 2 z ^{2}} - i \frac{2 z \frac{d}{d x} ( z ^{2} )}{1 + z ^{4} + 2 z ^{2}}

Schritte zur Lösung

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

(z + i)^{2} = z^{2} + 2 i z - 1

= \frac{\frac{d}{d x} ( z ^{2} )}{z ^{2} + 2 i z - 1}

Rationalisiere

\frac{\frac{d}{d x} ( z ^{2} )}{z ^{2} + 2 i z - 1} : \frac{( z ^{2} - 2 i z - 1 ) \frac{d}{d x} ( z ^{2} )}{z ^{4} + 2 z ^{2} + 1}

= \frac{\frac{d}{d x} ( z ^{2} ) ( z ^{2} - 2 i z - 1 )}{z ^{4} + 2 z ^{2} + 1}

Schreibe

\frac{\frac{d}{d x} ( z ^{2} ) ( z ^{2} - 1 - 2 i z )}{z ^{4} + 2 z ^{2} + 1}

in der Standard komplexen Form um:

\frac{z ^{2} \frac{d}{d x} ( z ^{2} ) - \frac{d}{d x} ( z ^{2} )}{z ^{4} + 2 z ^{2} + 1} - \frac{2 z \frac{d}{d x} ( z ^{2} )}{z ^{4} + 2 z ^{2} + 1} i

= \frac{z ^{2} \frac{d}{d x} ( z ^{2} ) - \frac{d}{d x} ( z ^{2} )}{z ^{4} + 2 z ^{2} + 1} - \frac{2 z \frac{d}{d x} ( z ^{2} )}{z ^{4} + 2 z ^{2} + 1} i

e x p an d (π + e) (π - e)

e x p an d \frac{30 a ^{2} + 6 a - 24}{5 a ( a - 4 )}

e x p an d x^{2} - 5 (- 2, - 1)

e x p an d \frac{0.0125 s ^{2} + s + 25}{( 0.5 s + 25 ) ( 0.025 s + 1 )}

e x p an d \frac{( 2 + 2 ) 2 - 2}{2}