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the number e has all sorts of amazing properties just as a review you can define it in terms of a limit the limit as n approaches infinity of 1 plus 1 over N to the nth power you could also define it as the limit as n approaches 0 of 1 plus n to the 1 over N power but what we're going to focus on this video is an amazing property of e and E has many many amazing properties but this is the one that's maybe most relevant to calculus and that's the notion that if I take the derivative with respect to X of e to the X that it is equal to drumroll it's equal to e to the X that to me is amazing let's just appreciate it for a second before we actually prove it this is part of the graph of y is equal to e to the X and so what this says is the derivative of e to the X for any X is equal to e to the X the slope of the tangent line at any point here is equal to the value of the function let's just appreciate that so right over here the value of the function is 1 and the slope of the tangent line is 1 here the value of the function is 2 and the slope of the tangent line is 2 here the value of the function is 4 and the slope of the tangent line is equal to 4 so I could just go on this is just another amazing thing about e and we'll see many more in calculus but let's now prove that this is actually true so let's just use our definition of a derivative so the derivative with respect to X of e to the X would be the limit of Delta X or as Delta X approaches 0 of e to the X plus Delta X minus e to the X all of that over all of that over Delta X now let's do some algebraic manipulation here to see if we can make some sense of it so this is going to be equal to the limit as Delta X approaches 0 of let's see what happens well I won't skip any steps here this is the same thing as e to the x times e to the Delta X this is just using our exponent properties here minus e to the X over Delta X just to be clear I rewrote this right over here as this right over here now I can factor out an e to the X and in fact because the e to the X is not affected as the as Delta X approaches zero I can factor the e to the X out of the entire limit so let's do that let's take it the e to the X out of the entire limit let's factor it let's factor it completely out it does not get affected by the Delta X so this is going to be equal to factor that e to the X out e to the X times the limit as Delta X approaches 0 of e to the Delta X minus 1 all of that over Delta X so now we're going to get a little bit fancy with our limits we're going to do what's known as a change of variable so I'm going to say let's see I don't know how to directly find this limit right over here but maybe I can simplify it and who knows maybe I could get it into one of these forms up here so what if I were to make the substitution and let me do it over here let's say I would make the substitution that n is equal to e to the Delta X minus 1 so what would this be if we were to solve for Delta X let's see we could add 1 to both sides n plus 1 is equal to e to the Delta X to solve for Delta X we could just take the natural logarithm log base e of both sides and we would get the natural log of n plus 1 is equal to Delta X and so we can make that substitution this could be replaced with this and what we have in the numerator over here can be replaced with and right over there and what would happen to the limit well as Delta X approaches 0 what does an approach so Delta X approaches 0 implies that n approaches but let's see is delta x approaches 0 this would be e to the 0 which is 1 minus 1 so it looks like n approaches 0 and you can look at it over here as n approaches 0 right over here natural log of 0 plus 1 natural log of 1 that is 0 so is each of them as Delta X approaches 0 and approaches 0 as n approaches 0 Delta X approaches 0 so then you can replace if you make the change of variable from Delta X to n you could still say as n approaches 0 so let me rewrite all of this that was the fanciest step in this entire thing that we're about to do so it's going to be e to the x times the limit since we changed our variable it's now going to be as n approaches 0 because it's Delta X approaches 0 and approaches 0 and vice versa and this numerator here we said hey that's going to be equal to N and over Delta X is now the natural log of n plus 1 natural log of n plus 1 now what does that do for us well what if we were divide the numerator and the denominator by n so let's multiply down here by 1 over N and let's multiply up here by 1 over N well our numerator is just going to be equal to 1 and what is our denominator equal well here we can just use our exponent properties all of this is going to be equal to this is going to be equal to we have our e to the X out front e to the X and then we have the limit as n approaches 0 our numerator is now 1 over now I'm going to rewrite this just using our logarithm properties if I have others right over here if I have a times the natural log of B this is the same thing as the natural log of B to the 8th power there's 8 power this is just natural log properties so what we have down here this would be the same thing as the natural log of n plus 1 and actually let me write it the other way around 1 plus n 1 plus n I just swapped these two to the 1 over N power so the natural log of that whole thing now you might be getting that tingly feel because something is starting to look familiar what I just constructed inside the logarithm here looks an awful lot like what we have right over here and limit as n approaches zero limit as n approaches zero and in fact we can use our limit properties this one isn't affected what's really affected is what's inside the logarithm so we could say that this is going to be equal to and we're approaching our drumroll e to the x times 1 over 1 over the natural log I'll do that blue color the natural log of I'll give myself some space the limit the limit as n approaches 0 of this business right over here which I could just write as 1 plus n to the 1 over N power so this is really interesting what is this what did I just what do I have there in the denominator with this limit what is this thing equal to well we already said that is a definition of e that is this right over here which is equal to this over here so this is equal to e so what is this all boil down to I think you see what this is going but this is fun we're downhill from here this is e to the x times 1 over the natural log the natural log of e well the natural log of e what power to have to raise easy to get to e well I just have to raise it to 1 so this gets us e to the X and we're done we've just proven that the derivative with respect to X of e to the X is indeed equal to e ZX that's an amazing finding that shows us one more dimension of the beauty of the number II