 In this video, consider the equation of the motion of some particle given as s equals 2t cube minus 5t squared plus 3t plus 4, where s is measured in centimeters and t is measured in seconds. We want to find the acceleration as a function of time and then figure out what the acceleration would be after two seconds. So the thing to remember when it comes to these physics problems is that if we take the derivative of position with this motion equation with respect to time, this is going to give us the velocity function. And if we take the second derivative, which is going to be the derivative of velocity, this is where the acceleration function comes into play here. The acceleration function is the change of velocity with respect to time and as velocity is the change of position with respect to time, acceleration is the second derivative. Now to compute the second derivative, we have to first compute the derivative. Go in order, right? It makes sense. For which, when we take the first derivative by using the linearity properties of the derivative and the power rule, we end up with 2 times 3, which is 6 times t squared. We lower the power by 1 minus 5 times 2, which is 10 times t. We lower the power by 1. And then the next one, we're going to get a plus 3 when we take the derivative. And when you take the derivative of constant, it just disappears. We don't even consider anymore. This is the first derivative. This gives us the velocity of the function. Then to find acceleration, we're going to take the derivative of the velocity function, which we observed already as the second derivative here of position. So just take the derivative of the previous function, for which we then are going to get 6 times 2, which is 12 times t. We lower the power by 1. Then the next one's going to be negative 10, because when you take the derivative of t, you just get back a 1. And when you take the derivative of 3, it becomes 0 right there. The derivative of constant is equal to 0. And so our acceleration function is going to be 12 t minus 10, 10 what? 12 minus t minus 10 what? What are the units here? Well, if we take the derivative, our position function, remember that was in centimeters, right? That means our velocity function is going to be measured in centimeters per second, because that's how we're measuring time. And so acceleration is going to be measured in centimeters per second squared. So our function negative 12 t minus 10, this will be measured in centimeters per second squared. If we want to figure out what's the acceleration after 2 seconds, we compute a of 2, which is going to give us 12 times 2 minus 10. 2 times 12 is 24, minus 10 is going to give us 14. And so the acceleration at time 2 will be 14 centimeters per second squared. We can see that with these shortcuts to computing the derivative. It's much easier to work with these physics problems where you have to calculate first and second derivatives to learn things about velocity and acceleration.