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AP.CALC:

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let's say that f of X is equal to the natural log of X and we want to figure out what the slope of the tangent line to the curve F is when X is equal to the number E when X is equal to the number E so here X is equal to the number E the point E comma 1 is on is on the curve F of E is 1 the natural log of E is 1 and I've drawn the slope of the tangent line and so I've drawn the tangent line and we need to figure out what the slope of it is or at least come up with an expression for it and I'm going to come up with an expression using both the formal definition and the alternate definition and that will allow us to compare them a little bit so let's think about first the formal definition so the formal definition wants us to find an expression for the derivative of our function at any X so let's say that this is some arbitrary X right over here this would be the point X comma f of X and let's say that this is let's call this X plus h X plus h so this distance right over here is going to be H this right over here is going to be the point X plus h f of X plus h now the whole the whole underlying idea of the formal definition of limits is to find the slope of the secant line between these two points the slope of the secant line between these two points and then take the limit as H approaches 0 as H gets closer and closer as H gets closer and closer this blue point is going to get closer and closer and closer to X and this point is going to approach it on the curve and the secant line is going to is going to become a better and better and better approximation of the tangent line of the tangent line at X so let's actually do that so what's the slope of the secant line what's the change in your vertical axis which is going to be f of X plus h minus f of X minus f of X over the change in your horizontal axis and that's X plus h minus X and we see here the difference is just H over H and we're going to take the limit of that we're going to take the limit of that as H approaches zero so in the case when f of X is the natural log of X this will reduce to the limit as H approaches zero f of X plus h is the natural log of X plus h minus minus the natural log of X all of that over H all of that over H so this right over here for our particular f of X this is this is equal to f prime of X so if we wanted to evaluate this when X is equal to e then everywhere where we see an X we just have to replace it with an e this is essentially expressing our derivative as a as a function of X it's kind of a crazy looking function of X you have a limit here and all of that but every place you see an X like any function definition you can replace it now with an e so we can see let me just let me just do that whoops I lost my screen here we go so we could write we could write F prime of e F prime of e is equal to the limit limit as H approaches zero of natural log we do in the same color so we can keep track of things natural log of E Plus H I'll just leave that blank for now minus the natural log of e all of that over H all of that over H so just like that this right over here if we evaluate this limit if we're able to and we actually can if we are able to evaluate this limit this would give us the slope of the tangent line when x equals e this is doing the formal definition now let's do the alternate definition the alternate definition if you want to find if you don't want to find a general function derivative expressed as a function of X like this and you just want to find the slope at a particular point the alternate definition kind of just gets straight to the point there so what they say is hey look let's imagine some other x-value here so let's imagine some other x-value this right over here is point is the point x comma well we could say f of X or we could even say the natural log of X X natural log of X what is the slope of the secant line between those two points what is the slope of the secant line between between those two points well it's going to be your change in Y values so it's going to be natural log of X natural log of X minus 1 minus 1 through that red color minus 1 over your change in X values that's X minus e x minus e X minus e so that's the slope of the secant line between those two points well what if you want to get the tangent line well let's just take the limit as X approaches e as X gets closer and closer and closer these points are going to get closer and closer and closer and the secant line is going to better approximate the tangent line so we're just going to take the limit as X approaches e so either one of this this is using the formal definition of a limit let me make clear that that H does not belong part of it so we can either do it during using the formal definition or or the alternate definition of the derivative

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