Exponential functions differentiation intro | Advanced derivatives | AP Calculus AB | Khan Academy

What I want to do in this video is explore taking the derivatives of exponential functions. So we've already seen that the derivative with respect to x of e to the x is equal to e to the x, which is a pretty amazing thing. One of the many things that makes e somewhat special is that when you have an exponential with your base right over here as e, the derivative of it, the slope at any point, is equal to the actual value of that actual function.

But now, let's start exploring when we have other bases. Can we somehow figure out what is the derivative with respect to x when we have a to the x, where a could be any number? Is there some way to figure this out, and maybe using our knowledge that the derivative of e to the x is e to the x? Well, can we somehow use a little bit of algebra and exponent properties to rewrite this so it does look like something with e as a base?

Well, you could view a as being equal to e. Let me write it this way: a is being equal to e to the natural log of a. Now, I want you, if this isn't obvious to you, I really want you to think about it. What is the natural log of a? The natural log of a is the power you need to raise e to, to get to a.

So if you actually raise e to that power, if you raise e to the power you need to raise e to get to a, well then you're just going to get to a. So really think about this; don't just accept this as a leap of faith. It should make sense to you and it just comes out of really what a logarithm is.

And so we can replace a with this whole expression here. We can exp—if a is the same thing as e to the natural log of a, well then this is going to be equal to the derivative with respect to x of e to the natural log of a, and then we're going to raise that to the x power. We're going to raise that to the x power.

And now, just using our exponent properties, this is going to be equal to the derivative with respect to x of—and I'll keep color coding it—if I raise something to an exponent then raise that to an exponent, that's the same thing as raising our original base to the product of those exponents. That's just a basic exponent property.

So that's going to be the same thing as e to the natural log of a times x power. And now we can use the chain rule to evaluate this derivative. So what we will do is we will first take the derivative of the outside function, so e to the natural log of a times x, with respect to the inside function, with respect to natural log of a times x.

And so this is going to be equal to e to the natural log of a times x. And then we take the derivative of that inside function with respect to x. Well, natural log of a might not immediately jump out to you, but that's just going to be a number. So that's just going to be—so times the derivative, if it was the derivative of 3x, it would just be 3.

If it's the derivative of natural log of a times x, it's just going to be natural log of a. And so this is going to give us the natural log of a times e to the natural log of a. And I'm going to write it like this: natural log of a to the x power. Well, we've already seen this.

Let me—this right over here is just a, so it all simplifies. It all simplifies to the natural log of a times a to the x, which is a pretty neat result. So if you're taking the derivative of e to the x, it's just going to be e to the x. If you're taking the derivative of a to the x, it's just going to be the natural log of a times a to the x.

And so we can now use this result to actually take the derivatives of these types of expressions with bases other than e. So if I want to find the derivative with respect to x of 8 times 3 to the x power, well what's that going to be? Well, that's just going to be 8 times, and then the derivative of this right over here is going to be, based on what we just saw, it's going to be the natural log of our base, natural log of 3, times 3 to the x.

So it's equal to 8 times natural log of 3 times 3 to the x.