Bacterial growth follows exponential growth. For example if a bacteria take 24 hours to divide, then at time = 0 there would be one, t=1 there would be 2, t=2 there would be 4, etc. What it means for an equation to be exponential in the real world is that the current number of the grower will have an impact on the growth. Thus the more things that can grow the more that will, and the longer the growth has been taking place then the more things there will be to grow. Thus, what this ends up meaning is that at any given moment the rate of change at time=t will be proportional to the the function itself. This is simply because as the function, i.e. the number of things growing, goes up, then there will be more things to which will grow, causing the rate of growth to go up. Vice Versa as well.
Question: Are there any other unique derivatives like e^x, that also apply to the real world?
karolmath 