 Now we're ready to look at a calculus-based version of the instantaneous velocity. We've already defined our instantaneous velocity to be the velocity at a specific point in time, given the symbol v. And we can compare this to the average velocity where we were working with a time span. Now graphically, when I was looking at the average velocity over a particular time span, I could connect two individual points and the slope of that line define both the time span and the displacement, giving us our average velocity. Well, if I reduce that time span but keep my same initial point, what I find is that as I reduce it, it changes the slope. But that slope is approaching a value, which is the tangent to the curve at that specific point in time. To put it into words, as the time span gets smaller, the estimate of that tangent gets better. And the exact value would happen when the time span goes to zero. Now mathematically, we represent that as the limit as delta t goes to zero of dx divided by dt. But in order to actually solve that equation, we need to use calculus. And it's called the derivative when we do that limit. Now this derivative equation here, a common mistake with beginning students, is how to think about what this equation is really saying. Shorthand will say v equals dx dt. But it's not the derivative of the position divided by the derivative of the time. Instead, it's the derivative of the position with respect to time. Another way this can be written to kind of highlight the fact that this derivative is a mathematical operation, and you're performing that operation on a specific variable of the position. Now we also want to remember that that position and that velocity are both functions of time. So when I take the derivative of the position with respect to time, I'm working with some function of time for the position, and I'm going to get some function of time for the velocity. Practically speaking, when I'm solving these kinds of problems, there's sort of three step process we're going through. You have to start with the position equation, so you have to have that function of time. Then you can take the derivative of that position equation with respect to time. And then you evaluate that at specific times to get the velocity at a specific time. Start off with the position equation. Here's one example we could use, and this is a simple polynomial type equation. Now a lot of textbooks will sort of strip out the units for you, so you can see very clearly that this is a function of time. Don't forget, however, even if you're working with this simplified form, that these are physical quantities and they really do have units associated with each and every one of those numbers. Once we've got our equation, then we can take the derivative with respect to time. So you can think of it as here's your derivative operation symbol, and you just plug in your function of position with respect to time. Now if you're well-practiced on your derivatives, then you would simply see that this derivative works out to be this equation. You may need to practice this a little bit, though, to remind yourself that that two meters is a constant, so the derivative disappears. The three t becomes just three. The four t squared becomes eight t. And notice that the units on these terms stick around. They don't get changed when you take the derivative. Now that you've got the equation for velocity as a function of time, to find specific velocities, you would just plug in the specific time that you're looking for. So we take our equation, but time becomes replaced with our variable two seconds. And as you multiply this all out, we would find that I've got 19 meters per second. Now here's where I say keeping your units in gives you an advantage, because you can double-check that yes, these really do have the correct units when I work everything out. And as I plug that in, it has the dimensions that I expect for a velocity. So that's your introduction to instantaneous velocity from a calculus-based perspective. You'll get plenty of practice working with this.