 Motion in one dimension, derivatives of polynomials. This tutorial is designed for students in general physics who are concurrently taking calculus and may not have had the rules for derivatives of polynomials by the time you need them in physics class to find the velocity in the acceleration. If you look in your typical calculus textbook for the polynomial rule of derivatives you'll probably find something like this. For a function of x equal to x to the power n, the derivative of that function of x with respect to x is equal to n times x to the power n minus 1. What does this mean and how does it help us find velocities in accelerations? To start with in physics we're not dealing with some generic function of a variable x. We're dealing with very specific functions of time. So we're first going to change our general rules here to use specifically a function of time, which is a power t to the n, and the derivatives with respect to time of those functions. Now the functions we're using in physics are position as a function of time, velocity as a function of time, and acceleration as a function of time. And that velocity as a function of time is equal to the derivative of the position with respect to time. And the acceleration as a function of time is a derivative of the velocity with respect to time. Now when you hear people pronounce these equations you might hear them use a shortened form such as v equals dx dt. But remember that that dx dt is the derivative with respect to time of x, the position function. You're always doing a mathematical operation when you're doing the derivative. So let's look at some examples. We'll take our general definitions and keep them down here at the bottom. If I have a position equation where the position as a function of time is equal to t cubed then my velocity as a function of time is the derivative with respect to time of that specific function t cubed. According to our polynomial rule that power three comes out front and my power is then reduced by one, three minus one and that simplifies down to three t squared. So if I start with a position t cubed my velocity is three t squared. Here's another example. Let's say I've got position as a function of time equal to t squared. The velocity is the derivative of t squared. By our polynomial rule the two comes out front. I've got two minus one which is just one and two t to the one is the same thing as two t. Now there's a couple of examples that might be a little more tricky. So for example what if I had five t squared? Well then my derivative is the derivative of five t squared and that five comes out front but I still have the same derivative of t squared. So instead of two t I now have ten t. So we can handle t squared, t cubed. You do a t to the fifth and a t to the sixth the same way and we can handle it with any arbitrary number out front even if it's a negative number. The negative number stays out front. Here's an example of just four t. Now to start with you're like well what's the power? But four t is the same thing as four t to the first power. So when we take our derivative of four t to the one four stays out front the power that comes down is just a one and my t is now t to the one minus one. So four t to the zero but t to the zero is the same thing as one so my derivative is just four. So if I start with a position of x equals four t I end with v equals four. What if I started with a position of just four? Constant number doesn't change with time. Well as we just talked about that's like four times one and four times one is the same thing as four t to the zero. So we're taking the derivative of four t to the zero and that power zero comes out front and then it doesn't matter what else you've got. So if you start with some constant you end with a velocity of zero. Now this makes sense when we remember what the derivative is really doing is measuring how our function is changing and a constant positions not changing. So my velocity is zero. If I always have the same position I'm not moving. Now that we have all of our little pieces let's try a more complicated example. Say a position as a function of time which has more than one term five minus four t squared plus two t cubed. Well then my velocity is the derivative with respect to time of that entire function. Luckily with our derivative rules we can break that up. Take the derivative with respect to time of five the derivative with respect to time of minus four t squared and the derivative with respect to time of two t cubed. Now we've seen all of these. The derivative of the constant just becomes zero for the one where you've got the squared number comes out front minus four times two t and then we've got two three t squared. Three t squared is the derivative of t cubed and all things simplifies down to a velocity of just minus eight t plus six t squared. We don't have to keep the zero in there and we multiply out our constants in front of the other numbers. Now let's say we wanted to take it just one step further. Let's take our position and our velocity that we just found and let's find the acceleration too. Well my acceleration as a function of time is the derivative with respect to time of my velocity equation so my minus eight t plus six t squared goes into here. Again we can break that derivative down term by term the derivative with respect to time of minus eight t the derivative with respect to time of six t squared. Minus eight t becomes just minus eight two comes out front and we multiply by the six which gives me twelve t. So if I have a position as a function of time which is a polynomial I can find the velocity as a function of time by taking the derivative once and I can find the acceleration as a function of time by taking the derivative a second time. That's why sometimes you'll hear the acceleration called the second derivative of the position. Hope this helps explain things. Good luck with all your physics problems.