erweitern (y-xy'(x))^3

Lösung

erweitern

(y - x y^{'} (x))^{3}

y^{3} - 3 x y^{2} y' (x) + 3 x^{2} y (y' (x))^{2} - x^{3} (y' (x))^{3}

Schritte zur Lösung

(y - x y^{'} (x))^{3}

Wende Formel für perfekte dritte Potenzen an:

(a - b)^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}

a = y, b = x y^{'} (x)

= y^{3} - 3 y^{2} x y^{^{'}} (x) + 3 y (x y^{^{'}} (x))^{2} - (x y^{^{'}} (x))^{3}

Vereinfache

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

= y^{3} - 3 x y^{2} y^{'} (x) + 3 x^{2} y (y^{'} (x))^{2} - x^{3} (y^{'} (x))^{3}

e x p an d \frac{4 x ^{5} + 2 x ^{4} - 4 x ^{3}}{( x + 1 ) ^{2}}

e x p an d - 2

e x p an d \frac{1}{3} (2 - x)^{3}

e x p an d \frac{s + 1}{s ^{2} ( s + 3 ) ( s + 2 )}

e x p an d \frac{( x + 1 ) ( x ^{5} - 4 x ^{4} + 4 x ^{3} - 6 x - 2 )}{x}