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let's say that this right over here is the graph of lowercase F of X that's lowercase F of X there and let's say that we have some other function capital f of X and if you were to take its derivative so capital F prime of X that's equal to lowercase F of X lowercase F of X so given that which of these which of these could be the graph of capital of capital f of X and I encourage you to pause this video and try to think about it on your own before we work through it well if if this curve is going to be the derivative of one of them that means at any for any x value its describing what the instantaneous rate of change or what the slope of the tangent line is of whichever one of these is the possible capital f of X so let's just look at a couple of things about here so what do we know about lowercase F of X what do we know which is the derivative of one of these well one thing we know it's it's always positive it has as we go to negative infinity at asymptotes towards zero but it's always positive so since this is describing the slope of one of these that means that the slope of one of these all of the candidates has to always be positive and if we look at this the slope of the tangent line here is indeed always positive slope of the tangent line here does look like it's positive every time we increase in X were increasing by Y here it's positive but here it's negative when we increase by X we decrease by Y so we can rule we can rule this one out now what else what else do we know well this is the derivative this is telling us the slope of the tangent line so for example when X is equal to when X is equal to negative 4 f of f of negative 4 is pretty close to zero it's pretty close to here it's a slightly slightly more than 0 so that tells us that the slope of the tangent line of capital f of X has to be pretty close to 0 when X is equal to negative 4 so let's see when X is equal to negative four here this slope the slope of the tangent line here isn't close to zero this actually looks closer to one so we could rule this one out over here when X is equal to negative four the slope of the tangent line and that actually does look pretty close to zero so I won't rule that one out and over here the slope of the tangent line when X is equal to negative four that also looks pretty close to zero so these are still both in the running so let's see how we can think of a difference so let's just pick another point what X is equal to what X is equal to 0 F of 0 looks like it's pretty close to 1 I don't know if it's exactly the one actually it looks almost exactly almost exactly equal to 1 so when capital f of so at capital f of 0 the slope of the tangent line needs to be pretty close to 1 so over here the slope of the tangent line when x is equal to 0 that looks smaller than 1 so this slope is definitely not 1 while over here when X is equal to 0 the slope of the tangent line does look the slope of the tangent line does look pretty pretty close pretty close to 1 so this right over here looks like the best candidate for capital for capital f of X so that one right over there I mean that is capital f of X and you might say hey these look very similar to each other in fact they look almost or actually they do look identical and you might remember from what you knew about differentiation actually these both look like the basic exponential function where I didn't ask you to figure out what the actual function was just what the possible antiderivative of this function would be this is the derivative lowercase F is the cat is the derivative of capital F or you could say that capital F is an antiderivative of lowercase F and when you just inspect this this looks like this the function both of these functions is our e to the X because the derivative of e to the X is e to the X