d/(dt)((x)(k(x^2-5x+4)))

Lösung

\frac{d}{d t} ((x) (k (x^{2} - 5 x + 4)))

k (3 x^{2} \frac{d x}{d t} - 10 x \frac{d x}{d t} + 4 \frac{d x}{d t})

Schritte zur Lösung

\frac{d}{d t} ((x) (k (x^{2} - 5 x + 4)))

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

= k \frac{d}{d t} (x (x^{2} - 5 x + 4))

Wende die Produktregel an:

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

f = x, g = x^{2} - 5 x + 4

= k (\frac{d x}{d t} (x^{2} - 5 x + 4) + \frac{d}{d t} (x^{2} - 5 x + 4) x)

\frac{d}{d t} (x^{2} - 5 x + 4) = 2 x \frac{d x}{d t} - 5 \frac{d x}{d t}

= k (\frac{d x}{d t} (x^{2} - 5 x + 4) + (2 x \frac{d x}{d t} - 5 \frac{d x}{d t}) x)

Multipliziere aus

\frac{d x}{d t} (x^{2} - 5 x + 4) + (2 x \frac{d x}{d t} - 5 \frac{d x}{d t}) x : 3 x^{2} \frac{d x}{d t} - 10 x \frac{d x}{d t} + 4 \frac{d x}{d t}

= k (3 x^{2} \frac{d x}{d t} - 10 x \frac{d x}{d t} + 4 \frac{d x}{d t})

e x p an d, 3 (4 - 2 i)

\frac{d}{d t} ((x) (- 2 x^{2} + 16 x))

e x p an d 121

e x p an d d x (6 x^{3} + 2 x + 1)

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