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Iamp;#39;ve introduced a few derivative formulas, but a really important one that I left out was exponentials. So here I want to talk about the derivatives of functions like 2 to the x, 7 to the x, and also to show why e to the x is arguably the most important of the exponentials. First of all, to get an intuition, letamp;#39;s just focus on the function 2 to the x. Letamp;#39;s think of that input as a time, t, maybe in days, and the output, 2 to the t, as a population size, perhaps of a particularly fertile band of pie creatures which doubles every single day. And actually, instead of population size, which grows in discrete little jumps with each new baby pie creature, maybe letamp;#39;s think of 2 to the t as the total mass of the population. I think that better reflects the continuity of this function, donamp;#39;t you? So for example, at time t equals 0, the total mass is 2 to the 0 equals 1, for the mass of one creature. At t equals 1 day, the population has grown to 2 to t
