So, it is indeed the case that the derivative is increasing when x is greater than zero, but that's not the justification for why h is increasing. For example, the derivative could be increasing while still being negative, in which case h would be decreasing. The appropriate justification is that h prime is positive, not that it's necessarily increasing, because you could be increasing and still not be positive. So, let's see. I would say that this does justify why h is increasing. When x is greater than zero, as the x values increase, the function values also increase. Well, that is a justification for why h is increasing, but that's not calculus-based, and no way are you using a derivative.