derivative of x^2(1-x^2)

Lösung

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

4 x^{3} - 6 x^{2} + 2 x

Schritte zur Lösung

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

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{2}, g = (1 - x)^{2}

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

\frac{d}{d x} (x^{2}) = 2 x

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

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

Vereinfache

2 x (1 - x)^{2} + (- 2 (1 - x)) x^{2} : 4 x^{3} - 6 x^{2} + 2 x

= 4 x^{3} - 6 x^{2} + 2 x