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## The chain rule: further practice

# Derivative of aˣ (for any positive base a)

AP.CALC:

FUN‑3 (EU)

, FUN‑3.C (LO)

, FUN‑3.C.1 (EK)

## Video transcript

- [Voiceover] 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 x, which is a pretty amazing thing. One of the many things that
makes e somewhat special. Though 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 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, 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, you could view a as being equal to e. Let me write it this way. Well all right, a as being equal to e to the natural log of a. Now 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 too 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. If a is the same thing as
e to the natural log of a, well then this is going to be, then this is going to be
equal to the derivative with respect to x of e to the natural log, I keep writing la (laughs), to the natural log of a and then we're going to
raise that to the xth power. We're going to raise that to the x power. And now this, just using
our exponent properties, this is going to be
equal to the derivative with respect to x of, and I'll just keep color-coding it. If I raise something to an exponent and 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, natural log of a times x power. 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, it 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
three x, it would just be three. If it's the derivative
of natural log 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. 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 eight times three to the x power, well what's that going to be? Well that's just going to be eight 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 three times three to the x. Times three to the x. So it's equal to eight natural log of three times three to the x. Times three to the x power.

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