d/(dt)((x)(3x+2y+4z))

Lösung

\frac{d}{d t} ((x) (3 x + 2 y + 4 z (t)))

6 x \frac{d x}{d t} + 2 y \frac{d x}{d t} + 4 z \frac{d x}{d t} + 2 x \frac{d y}{d t} + 4 x \frac{d z}{d t}

Schritte zur Lösung

\frac{d}{d t} ((x) (3 x + 2 y + 4 z))

Wende die Produktregel an:

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

f = x, g = 3 x + 2 y + 4 z

= \frac{d x}{d t} (3 x + 2 y + 4 z) + \frac{d}{d t} (3 x + 2 y + 4 z) x

\frac{d}{d t} (3 x + 2 y + 4 z) = 3 \frac{d x}{d t} + 2 \frac{d y}{d t} + 4 \frac{d z}{d t}

= \frac{d x}{d t} (3 x + 2 y + 4 z) + (3 \frac{d x}{d t} + 2 \frac{d y}{d t} + 4 \frac{d z}{d t}) x

Vereinfache

\frac{d x}{d t} (3 x + 2 y + 4 z) + (3 \frac{d x}{d t} + 2 \frac{d y}{d t} + 4 \frac{d z}{d t}) x : 6 x \frac{d x}{d t} + 2 y \frac{d x}{d t} + 4 z \frac{d x}{d t} + 2 x \frac{d y}{d t} + 4 x \frac{d z}{d t}

= 6 x \frac{d x}{d t} + 2 y \frac{d x}{d t} + 4 z \frac{d x}{d t} + 2 x \frac{d y}{d t} + 4 x \frac{d z}{d t}

s im pl i f y 4 (9 + x^{2}) + 5 (4 - x)

e x p an d (1 + sec (θ))^{2}

e x p an d \frac{( x + 5 ) ^{3}}{3}

e x p an d \frac{e ^{2 x} ( 5 e ^{3 x} - 2 )}{25}

s im pl i f y (\frac{1}{2} - 3 x) (\frac{1}{2} + 3 x)