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# Worked example: Derivative from limit expression

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

the alternate form of the derivative of the function f at a number a denoted by f prime of a is given by this stuff now this might look a little strange to you but if you really think about what it's saying it's really just taking the slope of the tangent line between a comma F of a so let's imagine some arbitrary function like this let's say that that is I'll just write that's our function f that's our function f and so you could have the point when X is equal to a this is our x-axis when X is equal to a this is the point a F of a you notice a f of a and then we could take the slope between that and some arbitrary point let's call that X so this is the point X f of X X f of X and notice this top the numerator right here this is just our change in the value of our function or you could view that as the change in the vertical axis so that would give you this distance right over here that's what we're doing up here in the numerator and then in the denominator we're finding the change in our horizontal values horizontal coordinates that let me do that in a different color that would give you that so the change in the horizontal that's this right over here and then they're trying to find the limit as X approaches a so as X gets closer and closer and closer and closer to a what's going to happen is is that when X is out here we have this secant line we have we're finding the slope of this secant line but as X gets closer and closer the secant lines better and better and better approximate the slope of the tangent line where the limit as X approaches a but does it quite equal a is going to be this is actually our definition of our derivative then or I guess the alternate form of the derivative definition and this would be the slope of the tangent line if it exists so with that all that out of the way let's try to answer their question with the alternative form of the derivatives in aid make sense of the following limit expression by identifying the function f and the number a so right here they want to find the slope of the tangent line at 5:00 here they wanted to find the slope of the tangent line at a so it's pretty clear that a is equal to five and that F of a F of a is equal to 125 now what about f of X f of X well here it's the limit of f of X minus f of a well here is the limit as X to the third minus 125 and this makes sense if f of X if f of X is equal to X to the third then it makes sense that f of five f of five is going to be five to the third is going to be 125 and we're also taking up here the limit as X approaches a here taking the limit as X approaches 5 so the derivative is the derivative of the function f of X is equal to X to the third let me write that down in the green color X to the third at the number x equals at the number a is equal to five and so we can imagine this let's try to actually graph it just so we can imagine it actually I'll do it out here or I have a little bit better contrast with the colors so let's say that is my y-axis let's say that this is my my x-axis I'm not going to quite draw it to scale let's say this right over here is the point 125 or Y this is when y equals 125 this is when X is equal to 5 so they're clearly not at the same scale but the function is going to look something it's going to look something like this we know what X to the third looks like it looks something something like something let me draw a little bit something like this we know it's going to look something like this so here our a is equal to 5 our a is equal to 5 this point right over here is is 5 comma 125 5 125 and then we're taking the slope between that point and an arbitrary x-value at an arbitrary or I should say an arbitrary other point on the curve so this right over here would be the point you call that X comma X to the third we know that f of X is equal to X to the third and let me make it clear this is a graph of y is equal to Y is equal to X to the third and so this expression right over here all of this this is the slope between these two points this is the slope between these two points and as we take the limit as X approaches 5 so right now this is our X as X gets closer and closer and closer to 5 as it gets closer and closer to 5 these secant lines are going to better and better approximate the slope of the tangent line at x equals at x equals 5 so the slope of the tangent line would look something it would be even more it would look something like something like that
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