Differentiate the Exponential from First Principles! | Calculus Tutorial! #calculus #maths

Ajouté : 2026-06-07

[00:00:00]Why is the derivative of e to the x equal to e to the x? Well, let's find out straight from the definition. Well, what is f prime of x? It is the limit as h goes to zero. It's the instantaneous rate of change of f at x. So, limit as h goes to zero of f of x plus h minus f of x divided by h. That's the limit we want to find out, and we can write this out as limit h goes to zero using the fact that f of x is e to the x of e to the x plus h minus e to the x divided by h.

[00:00:30]Now, I'm going to show you something really cool at the end as well, so you don't want to miss that. So, now we're going to write this out using exponent laws as limit h goes to zero of e to the x times e to the h minus e to the x divided by h. We can factor out an e to the x, okay? So, this is now the e to the x factors out, and because x is not a variable in the limit, it's a limit as h goes to zero, we can actually bring out the e to the x to the front and get limit h goes to zero of e to the h minus one divided by h. Now, this is the limit that we need to calculate to solve the problem, and I'm going to do that right up top. So, how do we find this limit?

[00:01:07]So, we cannot use L'Hopital's rule. Be very careful. L'Hopital's rule won't apply because this is the limit definition of the derivative of e to the x at x equals zero, this limit literally. So, applying L'Hopital's rule would be circular reasoning. We have to actually write out what e to the h is, which is it's an infinite series, sum n equals zero to infinity of h to the n divided by n factorial. Now, you can also write that out as one plus h plus h squared over two factorial and so on and so forth. Now, when you actually try to find out what is e to the h minus one, well, you're going to get rid of that one term, and you're going to be able to factor out a h because all the terms are going to have a h in them. So, it's going to be h times one plus h over two factorial plus h squared over three factorial and so on and so forth. Now, when you find the limit as h goes to zero, this is a very important point.

[00:02:01]You can actually do the limit term by term because this series converges uniformly. You have to be careful in math. You can't always do this. I'm telling you as a professional mathematician, but in this case as h goes to zero, you can do it term by term and everything goes to zero except the one. So, therefore, this limit is going to be one and the definition of the derivative is going to yield e to the x.

[00:02:22]Now, in fact, if you have any function whose derivative is equal to itself, it must actually pretty much be e to the x.

[00:02:29]And based on my lectures at Princeton University, I've given a rigorous proof in 5 minutes. It's a stunning trick.

[00:02:34]You're going to want to watch that. It's the first breakout video on my channel, one of the most popular videos on my channel. Check it out below. Click down below. Don't forget to smash that like button on this short for tons more fun math content that you can enjoy and benefit from and share with friends and family who are also learning and wanting to dive deeper into calculus.