d/(dt)(({r}(t))(sqrt(t)+t^3-t^2))

解

\frac{d}{d t} ((r (t)) (t + t^{3} - t^{2}))

\frac{d}{d t} (r (t)) (t + t^{3} - t^{2}) + (\frac{1}{2 t} + 3 t^{2} - 2 t) r (t)

解答ステップ

\frac{d}{d t} ((r (t)) (t + t^{3} - t^{2}))

積の法則を適用:

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

f = r (t), g = t + t^{3} - t^{2}

= \frac{d}{d t} (r (t)) (t + t^{3} - t^{2}) + \frac{d}{d t} (t + t^{3} - t^{2}) r (t)

\frac{d}{d t} (t + t^{3} - t^{2}) = \frac{1}{2 t} + 3 t^{2} - 2 t

= \frac{d}{d t} (r (t)) (t + t^{3} - t^{2}) + (\frac{1}{2 t} + 3 t^{2} - 2 t) r (t)