# Derivative intuition module

## Video transcript

This right here is our derivative intuition module. And it was a module that was contributed by Ben Eater, and it's a pretty interesting module, one, it's graphical and all the rest. But what's really cool about it is it really helps to conceptualize what a derivative is all about. And all it really is is the slope of the tangent line at any point. So in this exercise right over here, they've graphed f of x is equal to 2x to the third power, so this is 2x to the third power right over here. And what we need to do is move these orange dots up and down, so these orange dots right now show a line. So when I highlight on that orange dot, you saw this line down here became orange. And right now that orange dot is 0 because the slope of this line right over here is 0. But we see that that line right now is definitely not tangent, or the slope at that point in our curve 2x to the third, is definitely not 0. So what we need to do is we need to move this, so it becomes pretty close, as close as we can kind of eyeball it, to the slope of the tangent line. And so that looks about-- right about there looks about right. That looks pretty close to the slope of the curve at that point. And you could even see while I do it, up here, you see DDX of f of negative 2. So the derivative at f is equal to negative 2. So that's just saying that when the function is equal to negative 2, what is the derivative there? That's the slope. So by putting it all the way up here, I'm saying that the slope is equal to 24 at that point. So that's a pretty good approximation just eyeballing it like that. Now, let's do this point right over here. So it looks like the slope is decreasing a little bit, but it's still very positive. So the slope looks maybe it's close to around 14 or 15. And then over here, once again, our slope has decreased bit it's still reasonably positive. The slope looks like it's about 5, so the slope is a decreasing. Right over here the slope does look like it's about 0 right over here. This looks like an inflection point actually. So our slope is 0 right there. And you see the derivative at f is 0 is 0, which just means this is fancy notation to say that the slope of the tangent line, or you could say the slope of the curve at that point, is 0. It's flat there, and you see it really is flat. And then you go over here, and we want to find the derivative at f of 1. And so once again, it looks like the slope is now increasing. So this is interesting. Up here, the slope was very positive, but it was decreasing as we went here. Notice, it's getting flatter and flatter. Then it goes to 0, and now the slope starts increasing again. So at this point, we want to make it tangent. And so now notice the slope has gone up relative to when we were at 1. So that looks like the slope right over there. And then we do the slope over there. And if we get pretty close, it'll actually draw the curve of the derivative. So there you go. And so what's neat about here, we didn't find the derivative at every point in the curve, we found it at just add the special points that we were able to move these orange dots for. But since we got the orange dots close enough to the actual derivatives, just by eyeballing it, just trying to find the-- trying to look at what the slope of this tangent line is, it said, OK, you've done it right, and it drew the whole slope of the derivative. And what this entire derivative is saying is at any point, this is the slope of the tangent line. So even though we didn't move a dot around at negative 0.5, if you go right up here, that looks like it's around-- this is 2 and 1/2, so this would be a little bit over 1. It looks like that would be the slope of the tangent line right over there. Or if we went all the way to negative 2.25, the slope of the tangent line looks like it's approximately 30 right over there. So once you have the curve, you now have the slope of the tangent line, or you have the slope of the curve, at every point that we see over here. So this is just, I thought, a pretty neat exercise.