erweitern (x(x+h)^3-(x^3))/h

Lösung

erweitern

\frac{x ( x + h ) ^{3} - ( x ^{3} )}{h}

\frac{x ^{4}}{h} + 3 x^{3} + 3 x^{2} h + x h^{2} - \frac{x ^{3}}{h}

Schritte zur Lösung

\frac{x ( x + h ) ^{3} - ( x ^{3} )}{h}

Entferne die Klammern:

(a) = a

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

(x + h)^{3} = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}

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

Multipliziere aus

x (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) - x^{3} : x^{4} + 3 x^{3} h + 3 x^{2} h^{2} + x h^{3} - x^{3}

= \frac{x ^{4} + 3 x ^{3} h + 3 x ^{2} h ^{2} + x h ^{3} - x ^{3}}{h}

Wende Bruchregel an:

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

\frac{x ^{4} + 3 x ^{3} h + 3 x ^{2} h ^{2} + x h ^{3} - x ^{3}}{h} = \frac{x ^{4}}{h} + \frac{3 x ^{3} h}{h} + \frac{3 x ^{2} h ^{2}}{h} + \frac{x h ^{3}}{h} - \frac{x ^{3}}{h}

= \frac{x ^{4}}{h} + \frac{3 x ^{3} h}{h} + \frac{3 x ^{2} h ^{2}}{h} + \frac{x h ^{3}}{h} - \frac{x ^{3}}{h}

Streiche

\frac{3 x ^{3} h}{h} : 3 x^{3}

= \frac{x ^{4}}{h} + 3 x^{3} + \frac{3 x ^{2} h ^{2}}{h} + \frac{x h ^{3}}{h} - \frac{x ^{3}}{h}

Streiche

\frac{3 x ^{2} h ^{2}}{h} : 3 x^{2} h

= \frac{x ^{4}}{h} + 3 x^{3} + 3 x^{2} h + \frac{x h ^{3}}{h} - \frac{x ^{3}}{h}

Streiche

\frac{x h ^{3}}{h} : x h^{2}

= \frac{x ^{4}}{h} + 3 x^{3} + 3 x^{2} h + x h^{2} - \frac{x ^{3}}{h}

e x p an d \frac{5 ( 2 x + 9 )}{12}

e x p an d \frac{1}{2 ( 1 + 2 h ) [ 6 + 6 h ]}

e x p an d v (0) f (0, - \frac{1}{39})

e x p an d \frac{1}{y ( 2 - y ) ( 4 - y )}

e x p an d \frac{( x + 1 ) ^{2}}{( x + 2 ) ^{4}}