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# Formal and alternate form of the derivative

AP.CALC:
CHA‑2 (EU)
,
CHA‑2.B (LO)
,
CHA‑2.B.2 (EK)
,
CHA‑2.B.3 (EK)
,
CHA‑2.B.4 (EK)

## Video transcript

so let's think about how we could find the slope of the tangent line to this curve right over here so what I've drawn in red at the point x equals a and we've already seen this with the definition of the derivative we could try to find a general function that gives us the slope of the tangent line at any point so let's say we have some arbitrary point let me define some arbitrary point X right over here then this would be the point X comma f of X and then we could take some x + H so let's say that this right over here is the point X plus h and so this point would be X plus h f of X plus h we can find the slope of the secant line that goes between these two points so that would be your change in your vertical which would be f of X plus h f of X plus h minus f of X minus f of X over the change in the horizontal which would be X plus h minus X X plus h minus X minus X and these two x's cancel so this would be the slope of this secant line and then if we want to find the slope of the tangent line at X the slope of the tangent line at X we would just take the limit the limit of this expression as H approaches 0 as H approaches 0 this point moves towards X and that slope of the secant line between these two is going to approximate the slope of the tangent line at X and so this right over here this we would say is equal to this is equal to F prime of X this is still this is a function of X you give me an X you give me an arbitrary X that where the derivative is defined I'm going to plug it into this whatever this ends up being it might be some nice clean algebraic expression that I'm going to give you a number so for example if you wanted to find you you could calculate this somehow or you could even leave it in this form and then if you wanted F prime of a if you wanted F prime of a you would just substitute a into your function definition and you would say well that's going to be the limit as H approaches zero of every place you see an X replace it with an A f of I'll stay in this color for now blank plus h minus minus F minus F of blank minus F of blank all of that over all of that over H and I left those blanks so I could write the a I could write the a in red notice every place where I had an X before it's now an A so this is the derivative evaluated at a so this is one way to find the slope of the tangent line when x equals a another way and this is often used as the alternate form of the derivative would be to do it directly so this is the point a comma F of a let's just take another arbitrary point some here someplace so let's say this is the value X this point right over here on the function would be X comma f of X and so what's the slope of the secant line between these two points what would be change in the vertical which would be f of X minus f of a minus F of a over a change in the horizontal over X minus a so let me do that purple color over X minus a now how can we get a better and better approximation for the slope of the tangent line here well we could take the limit as X approaches a as X gets closer and closer and closer to a the secant line slope is going to better and better and better approximate the slope of the tangent line this tangent line that I have in red here so we would want to take the limit the limit as as X approaches a here either way we're doing a very similar we're doing the exact same thing we're finding that we're taking this we're have an expression for the slope of a secant line and then we're bringing those those X values of those points closer and closer together closer and closer together so the slopes of those secant lines better and better and better approximate that slope of the tangent line and at limit it does become the slope of the tangent line that is the derivative of that's the definition of the derivative so this is the clot that kind of the more standard definition of a derivative it would give you your derivative as a function of X and then you can then input your X that you want your the the particular value of x or you could use the alternate form of the derivative if you know that hey look I'm just looking to find the derivative exactly at a I don't need a general function of F then you could do this but they're doing the same thing
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