Proof: The derivative of __ is __ | Advanced derivatives | AP Calculus AB | Khan Academy

The number e has all sorts of amazing properties. Just as a review, you can define it in terms of a limit: the limit as n approaches infinity of 1 + 1/n to the nth power. You could also define it as the limit as n approaches zero of 1 + n to the 1/nth power.

But what we're going to focus on in this video is an amazing property of e. E has many, many amazing properties, but this is the one that's made most relevant to calculus. That's the notion that if I take the derivative with respect to x of e to the x, that it is equal to, drum roll, it's equal to e to the x. That, to me, is amazing!

Let's just appreciate it for a second before we actually prove it. This is part of the graph of y is equal to e to the x, and so what this says is the derivative of e to the x for any x is equal to e to the x. The slope of the tangent line at any point here is equal to the value of the function. Let's just appreciate that!

So right over here, the value of the function is one, and the slope of the tangent line is one. Here, the value of the function is two, and the slope of the tangent line is two. Here, the value of the function is four, and the slope of the tangent line is equal to four. So I could just go on. This is just another amazing thing about e, and we'll see many more in calculus.

But let's now prove that this is actually true. So let's just use our definition of a derivative. The derivative with respect to x of e to the x would be the limit of delta x as delta x approaches zero of e to the x + delta x minus e to the x all of that over delta x.

Now let's do some algebraic manipulation here to see if we can make some sense of it. So this is going to be equal to the limit as delta x approaches zero of, let's see what happens! Well, I won't skip any steps here. This is the same thing as e to the x times e to the delta x—this is just using our exponent properties—minus e to the x over delta x.

Just to be clear, I rewrote this right over here as this right over here. Now I can factor out an e to the x. In fact, because e to the x is not affected as delta x approaches zero, I can factor the e to the x out of the entire limit.

So let's do that. Let's take the e to the x out of the entire limit; let's factor it completely out. It does not get affected by the delta x, so this is going to be equal to: factor that e to the x out, e to the x times the limit as delta x approaches zero of e to the delta x - 1 all of that over delta x.

So now we're going to get a little bit fancy with our limits. We're going to do what's known as a change of variable. So I'm going to say, let's see, I don't know how to directly find this limit right over here, but maybe I can simplify it, and who knows, maybe I could get it into one of these forms up here.

So what if I were to make the substitution—and let me do it over here—let's say I would make the substitution that n is equal to e to the delta x minus one. So what would this be? If we were to solve for delta x, let's see, we could add one to both sides: n + 1 is equal to e to the delta x. To solve for delta x, we could just take the natural logarithm (log base e) of both sides, and we would get the natural log of n + 1 is equal to delta x.

So we can make that substitution. This could be replaced with this, and what we have in the numerator over here can be replaced with n right over there. And what would happen to the limit? Well, as delta x approaches zero, what does n approach?

So delta x approaches zero implies that n approaches what? Let's see, as delta x approaches zero, this would be e to the 0, which is 1 - 1, so it looks like n approaches 0. And you could look at it over here: as n approaches 0 right over here, natural log of 0 + 1, natural log of 1, that is 0.

So, as each of them, as delta x approaches zero, n approaches zero. As n approaches 0, delta x approaches zero. So then you can replace—if you make the change of variable from delta x to n—you could still say as n approaches zero.

So let me rewrite all of this. That was the fanciest step in this entire thing that we're about to do. So this is going to be e to the x times the limit—since we changed our variable, it's now going to be as n approaches zero—because as delta x approaches zero, n approaches zero and vice versa.

This numerator here, we said, "Hey, that's going to be equal to n." n over delta x is now the natural log of n + 1. Now, what does that do for us? Well, what if we were to divide the numerator and the denominator by n?

So let's multiply down here by 1/n and let's multiply up here by 1/n. Well, our numerator is just going to be equal to one, and what does our denominator equal? Well, here we can just use our exponent properties—this is going to be equal to, we have our e to the x out front, e to the x, and then we have the limit as n approaches zero our numerator is now one over now I'm going to rewrite this just using our logarithm properties.

If I have a times the natural log of b, this is the same thing as the natural log of b to the a power, the e to the power. This is just natural log properties. So what we have down here, this would be the same thing as the natural log of n + 1. And actually, let me write it the other way around: 1 + n—I just swapped these two—to the 1/n power, so the natural log of that whole thing.

Now, you might be getting that tingly feeling because something is starting to look familiar. What I just constructed inside the logarithm here looks an awful lot like what we have right over here—the limit as n approaches zero, limit as n approaches zero.

In fact, we can use our limit properties. This one isn't affected; what's really affected is what's inside the logarithm. So we could say that this is going to be equal to—and we're approaching our drumroll—e to the x times one over the natural log.

I'll do that in blue color—the natural log of, I give myself some space, the limit as n approaches zero of this business right over here, which I could just write as 1 + n to the 1/n power. So this is really interesting.

What is this? What did I just—I have there in the denominator with this limit. What is this thing equal to? Well, we already said that is a definition of e! That is this right over here, which is equal to this over here, so this is equal to e.

So what does this all boil down to? I think you see where this is going, but this is fun; we're downhill from here! This is e to the x times one over the natural log—the natural log of e. Well, the natural log of e, what power do I have to raise e to get to e? Well, I just have to raise it to one!

So this gets us e to the x, and we're done! We've just proven that the derivative with respect to x of e to the x is indeed equal to e to the x. That's an amazing finding that shows us one more dimension of the beauty of the number e.