 Now we're ready to talk about the calculus-based version of instantaneous acceleration. Acceleration at a specific point in time is what we're talking about here. And the instantaneous acceleration is the default form. So when they just say acceleration, they mean the instantaneous. And we want to compare this to what we saw with average acceleration, rather than a specific point in time that involved a time span. Now we've already seen that we have to do these calculus limits. And we saw this back when we started doing velocity. Now we're looking at acceleration. So let's compare what we saw with velocity with what we've got now. And as a reminder, the average velocity dealt with the time period. When I take the limit as the delta t goes to zero, that's what gives me my instantaneous velocity. And in calculus, that's the derivative. And so in the velocity, it was the derivative of the position with respect to time. I've got the same sort of thing when it comes to acceleration, except for now it's the change in the velocity or the derivative of the velocity with respect to time. But it still has to do with taking this limit as delta t goes to zero. Now for our equation then, the acceleration is the dv dt, is how we sort of pronounce it in shorthand. But that's really supposed to be the derivative of the velocity with respect to time. And so that's why we sometimes write it out in terms of putting the derivative with respect to time as the operation, which is acting on the variable of the velocity. If I keep this form in mind, I can learn a little bit more. So here I've got the derivative with respect to time of the velocity. But if you remember, the velocity was the derivative with respect to time of the position. So we've got two different derivatives that are happening here. And this means we're doing the second derivative of the position with respect to time. We can write this in two ways. One is to note that it's the derivative operation, which is squared because I'm doing it twice. Or in shorthand, it's d squared x dt squared. But it's really better to start pronouncing this as the second derivative of the position with respect to time. Now the process, it's the same sort of process we used in velocity, but now we've got two different paths we can go. If you're starting with a velocity function, then you can take the derivative of that velocity function and then evaluate that derivative at a specific time to find your acceleration at a particular time. If you start with the position function, you have to take the derivative twice and then evaluate it at a specific time. So let's take an example of this taking the derivative twice. So we're going to start with a position function. And like we did with our velocity examples, we're going to start with a polynomial. Again, I've included the units on each of these terms. You'll see in some textbooks that they shorthand this to just 2 plus 3t plus 4t squared plus 5t cubed. But remember that positions actually are physical quantities, and so each of these numbers has a unit associated with them. Well, when we take the velocity, we're finding the first derivative. When I take the first derivative of this equation, I get this. Again, if you're not really familiar with your derivatives, you're going to have to practice this, and you might want to go look at the separate video describing how to take derivatives of polynomials. Once we've gotten this velocity equation, which is the first derivative of the position, you then want to take the second derivative. So we've taken the position, and the first time we do the derivative, that's giving us the velocity. The second time we do the derivative, that's giving us the acceleration. And notice that we reduce the number of terms each time because the constant terms fall off or go to zero each time we take a derivative. Now, if I wanted to find the acceleration at a specific point in time, I'd just have to fill that time in for t, multiply out and add in. This gives you just the basic introduction to the calc-based view of acceleration. You're going to have to practice that, and your teacher's going to show you some more examples.