erweitern (((1-x)^3)'2x^2-(1-x)^3(2x^2)')/(2x^4)

Lösung

erweitern

\frac{( ( 1 - x ) ^{3} ) ^{'} \cdot 2 x ^{2} - ( 1 - x ) ^{3} \cdot ( 2 x ^{2} ) ^{'}}{2 x ^{4}}

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

Schritte zur Lösung

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

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

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

Multipliziere aus

((1 - x)^{3})^{'} 2 x^{2} - (1 - 3 x + 3 x^{2} - x^{3}) (2 x^{2})^{'} : ((1 - x)^{3})^{'} 2 x^{2} - (2 x^{2})^{'} + 3 x (2 x^{2})^{'} - 3 x^{2} (2 x^{2})^{'} + x^{3} (2 x^{2})^{'}

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

Wende Bruchregel an:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

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

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

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

\frac{3 x ( 2 x ^{2} ) ^{'}}{2 x ^{4}} = \frac{3 ( 2 x ^{2} ) ^{'}}{2 x ^{3}}

\frac{3 x ^{2} ( 2 x ^{2} ) ^{'}}{2 x ^{4}} = \frac{3 ( 2 x ^{2} ) ^{'}}{2 x ^{2}}

\frac{x ^{3} ( 2 x ^{2} ) ^{'}}{2 x ^{4}} = \frac{( 2 x ^{2} ) ^{'}}{2 x}

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

e x p an d \frac{32 x ^{2} + 288 x + 640}{8 x + 40}

e x p an d \frac{( x + 2 ) ( x + 4 )}{x + 3}

e x p an d \frac{1}{( T j x + 1 ) ^{2}}

e x p an d \frac{3 ( x + 4 )}{x}

e x p an d 5.24 (300)^{0.49}