We're going to raise that to the x power. If a is the same thing asĮ to the natural log of a, well then this is going to be, then this is going to beĮqual 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 Don't just accept this as a leap of faith. 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. 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. Now if this isn't obvious to you, I really want you to think about it. Well all right, a as being equal to e to the natural log of a. Something with e as a base? Well, you could view a, you could view a as being equal to e. Well can we somehow useĪ little bit of algebra and exponent properties to rewrite this so it does look like Knowledge that the derivative of e to the x, is e to the x. Where a could be any number? Is there some way to figure this out? And maybe using our What is the derivative, what is the derivative with respect to x when we have a to the x, The slope at any point, is equal to the value Though when you have an exponential with your base right over here as e, the derivative of it, 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. Want to do in this video is explore taking the derivatives