derivative of ((x(x-1)/(2x^2(x+1)))^3)

Lösung

\frac{d}{d x} ((\frac{x ( x - 1 )}{2 x ^{2} ( x + 1 )})^{3})

\frac{3 ( x - 1 ) ^{2} ( - x ^{2} + 2 x + 1 )}{8 x ^{4} ( x + 1 ) ^{4}}

Schritte zur Lösung

\frac{d}{d x} ((\frac{x ( x - 1 )}{2 x ^{2} ( x + 1 )})^{3})

Wende die Kettenregel an:

3 (\frac{x ( x - 1 )}{2 x ^{2} ( x + 1 )})^{2} \frac{d}{d x} (\frac{x ( x - 1 )}{2 x ^{2} ( x + 1 )})

= 3 (\frac{x ( x - 1 )}{2 x ^{2} ( x + 1 )})^{2} \frac{d}{d x} (\frac{x ( x - 1 )}{2 x ^{2} ( x + 1 )})

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

= 3 (\frac{x ( x - 1 )}{2 x ^{2} ( x + 1 )})^{2} \frac{- x ^{2} + 2 x + 1}{2 x ^{2} ( x + 1 ) ^{2}}

Vereinfache

3 (\frac{x ( x - 1 )}{2 x ^{2} ( x + 1 )})^{2} \frac{- x ^{2} + 2 x + 1}{2 x ^{2} ( x + 1 ) ^{2}} : \frac{3 ( x - 1 ) ^{2} ( - x ^{2} + 2 x + 1 )}{8 x ^{4} ( x + 1 ) ^{4}}

= \frac{3 ( x - 1 ) ^{2} ( - x ^{2} + 2 x + 1 )}{8 x ^{4} ( x + 1 ) ^{4}}