interpretation of the derivative with respect to time in the Ricci flow equation

Question

I would like to geometrically understand what $ \frac{d}{dt} g (x, t) $ means, since I know that the metric is a tensor of rank two and that it can be derived but not in this way in all the books and Articles that I have read do not explain it either, so what does it represent for you?

Answer 1

That's just how fast $g$ is changing with time.