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

解

\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}

解答ステップ

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

積の法則を適用:

(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

簡素化

\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}