Can instantaneous rate of change be negative?
Most certainly! When the instantaneous rate of change of a function at a given point is negative, it simply means that the function is decreasing at that point. As an example, given a function of the form
This can also be seen in physics. In physics, when your velocity, or your rate of change of position, is positive, that means that you are moving in the 'positive' direction (such as towards the right on a number line). When your velocity is negative, it means that you are moving in the 'negative' direction (such as towards the left on the number line). Further, if your acceleration, your rate of change of velocity, is positive, it means that your velocity is increasing, and if your acceleration is negative, then your velocity is decreasing.