If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Derivative of aˣ (for any positive base a)

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.C (LO)
,
FUN‑3.C.1 (EK)

## Video transcript

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 that when you have an exponential with your base right over here is e the derivative of it the slope at any point is equal to the actual is equal to the value of that actual function but now let's see 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 alright a is being equal to e to the natural log of a now I want you to if this isn't is this isn't obvious 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 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 to get to a well then you're just going to get to a so really think about this if 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 of a is the same thing as e to the natural log of a well then this is going to be then then this is going to be equal to the derivative with respect to X of e e to the natural log I keep writing la 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 this 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 properties 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 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 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 x 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 the natural law 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 times 3 to the X so it's equal to 8 natural logs of 3 x times 3 to the x times 3 to the X power
AP® is a registered trademark of the College Board, which has not reviewed this resource.