 Hello and welcome to a screencast about sketching a derivative graph. The question that we're looking at today is given the graph of a function, how do we sketch the graph of its derivative? Okay, so sketch an approximate graph of the derivative given the following function and the keyword here is approximate. We're not going to mold anything that's going to be exact, but we're going to have a good idea of what's happening. Okay, so as we read this graph from left to right, you can tell that the graph is increasing until we get to about this point up here. And if we were to draw a tangent line in, that would be horizontal. So we know that on the derivative, that's going to then be a zero on our derivative graph. Okay, now our function's going to be decreasing until we get about two, I'd say about that point right there. And again, that's going to represent a zero on the derivative. And then we're increasing until we get to about right here. And again, we'd have another horizontal tangent. So that's going to represent then a zero on this graph. Okay, so let's go back then and look at the pieces in between. So this piece right here to the left of our first zero is, our first horizontal tangent is going to be increasing. So we know that increasing means it's got a positive slope. So that means that our derivative is going to have to be something that's positive. So that means above the axis. Now again, you notice that right in here, the slope is really steep. We're over here, it's not quite so steep. So we know it's going to be something that's then going to have to look something like this. Because here's the derivative's bigger, here the derivative's smaller. Okay, now between these two points, okay? So between these two horizontal tangents, we know that the graph is decreasing. And again, you can kind of go through an eyeball the rates that it's decreasing at. But we know that we're going to have to have something that's negative on our derivative, and we're going to have to connect it up with this point over here. So we could say the graph's going to do something kind of like that. And pretend like this is actually fairly smooth in here. It's got a little bit out of control. There we go, that's better. Okay, then now from this horizontal tangent to this horizontal tangent, we have something that's increasing. And again, if you were to look at the rates of increase, you know that they're going to be changing, they're not going to be constant. So we're going to have to then connect this horizontal tangent to this horizontal tangent. So that's going to have to look something like this. It's a little bit more smooth. And then after this last horizontal tangent, we know the graph is decreasing. So it's going to have to be something that's negative. Okay, now how do we know where these peaks or valleys are? Honestly, at this point, we don't. So until we know more information, this is a good approximate sketch of the derivative. Okay, looking at a graph that's similar, but yet different enough. So again, we want an approximate graph of the derivative. But you notice this graph is not so curvy. This graph is more choppy. So here, we have a graph that is increasing right away until we hit this point up here at the top. But you notice that it's increasing at the exact same slope. Okay, now I don't have any grid lines on here, but we do know that it's going to be flat, it's going to be a constant slope. So we know we're going to have to have a flat or a constant derivative. So I'm going to go ahead and put that, let's just say right in here. So where this is at, like I said, we don't know what the grid lines are, so we don't know exactly what we're doing as far as the scale goes. So as long as you make it flat and consistent, then you know you're correct. And if we eyeball over here to this other part, we know that it's going to have the exact same slope. And again, it's going to be constant. So I'm going to fill that in here. Okay, then I could say that then the pieces on the downward parts of these slants are the same slope, but it's exactly the negative of that slope. Because these are definitely decreasing. So I'm going to skip down here, and then we've got that here, and then we've got that here. Now, what do we do with these endpoints? That's going to be definitely the kicker. So what is happening up here at this peak, this valley, and this peak? Now the endpoints we really can't do anything about, because we don't know what's happening after those. So those we can just kind of leave alone. But at these two peaks in this valley down here, what would you say is happening with the graph? Well, the fact that it's at a sharp corner, you guys have learned or will be learning soon, that that means our function is undefined, our derivative is undefined there. So I'm going to want to put an open circle here, and an open circle here. Open circle here, and an open circle here. Open circle here, and an open circle here. Okay, now if you did not put in those open circles, you would not be indicating that that derivative is undefined. And that's a very important point, because how can the derivative jump from an immediate point where it's positive to negative, without having some sort of a jump in the graph? So that's why you want to make sure that you show those points where the derivative is not defined by putting in those open circles. So again, our derivative is going to be this kind of step piecewise function here drawn in red. Thank you for watching.