Derivatives of exponentials | Chapter 5, Essence of calculus
What is e? And why are exponentials proportional to their own derivatives? Full series: http://3b1b.co/calculus Series like this one are funded largely by the ...
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I've introduced a few derivative formulas but a really important one that Ieft out was exponentials. So here, I want to talk about the derivatives of functions like Two to the x, seven 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, let's just focus on the function two to the x. And let'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 pi creatures which doubles every single day. And actually, instead of population size, which grows in discrete little jumps with each new baby pi creature, maybe let's think of 2 to the t as the total mass of the population. I think that better reflects th...
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I've introduced a few derivative formulas but a really important one that Ieft out was exponentials. So here, I want to talk about the derivatives of functions like Two to the x, seven 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, let's just focus on the function two to the x. And let'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 pi creatures which doubles every single day. And actually, instead of population size, which grows in discrete little jumps with each new baby pi creature, maybe let's think of 2 to the t as the total mass of the population. I think that better reflects th...
