Derivative of __ | Advanced derivatives | AP Calculus AB | Khan Academy

What we have right over here is the graph of ( y ) is equal to ( e^x ). What we're going to know by the end of this video is one of the most fascinating ideas in calculus, and once again, it reinforces the idea that ( e ) is really this somewhat magical number.

So, we're going to do a little bit of an exploration. Let's just pick some points on this curve of ( y ) is equal to ( e^x ) and think about what the slope of the tangent line is, or what the derivative looks like.

Let's say when ( y ) is equal to 1, or when ( e^x ) is equal to 1. This is the case when ( x ) is equal to zero. Well, the slope of the tangent line looks like it is 1, which is curious because that's exactly the value of the function at that point.

What about when ( e^x ) is equal to 2, right over here? Well, here let me do that another color. The slope of the tangent line sure looks pretty close—sure looks pretty close to 2.

What about when ( e^x ) is equal to 5? Well, the slope of the tangent line here sure does look pretty close—sure does look pretty close to 5.

So, just eyeballing it, is it the case that the slope of the tangent line of ( e^x ) is the same thing as ( e^x )? I will tell you, and this is an amazing thing, that that is indeed true. If I have some function ( f(x) ) that is equal to ( e^x ), and if I were to take the derivative of this, this is going to be equal to ( e^x ) as well.

Another way of saying it, the derivative with respect to ( x ) of ( e^x ) is equal to ( e^x ). That is an amazing thing. In previous lessons or courses, you've learned about ways to define ( e ), and this could be a new one.

( e ) is the number that, where if you take that number to the power ( x )—if you define a function or expression as ( e^x )—it's that number where if you take the derivative of that, it's still going to be ( e^x ).

What you're looking at here is a curve where the value, the ( y ) value at any point, is the same as the slope of the tangent line. If that doesn't strike you as mysterious and magical and amazing just yet, it will. Maybe tonight you'll wake up in the middle of the night and you'll realize just what's going on.

Now, some of you might be saying, "Okay, this is cool, you're telling me this, but how do I know it's true?" In another video, we will do the proof.