so the last video for this section will be a pretty short one comparatively um and it covers something that we previously went over so if s of T is a position function then S Prime of T is a velocity function now in this example I'm just going to use feet in seconds but it could be miles and hours meters per second inches per second centimeters per minute it's just the distance traveled as a function of time and so if I were to look at the distance here right that is the change in feet over the change in seconds which is feet per second the slope of s of T tells you the Velocity in feet per second at that moment so if we had a position function in number of feet as a function of seconds then the slope would tell you the feet per second traveled at any given moment right if I said you know how fast we're going at two seconds the slope here would tell you the velocity at that point which says the derivative of your position function is a velocities function so given s of T the derivative is S Prime but because it's a velocity function sometimes they refer to that as V of T and so this would tell us the output in velocity which would be the feet per second traveled as a function of seconds so for any time you could plug it into the velocity function and you would find your velocity the feet per second you were traveling at that time now if s prime or V of T is a velocity function then the derivative of this function also has meaning which could be thought of as s double prime or V Prime but if we have a velocity function V then the derivative V Prime is an acceleration function right just like this is the change in feet over seconds which becomes a velocity the change in velocity whether you speed up or speed down that is acceleration right when you change velocity if you increase speed you accelerate if you decrease speed you decelerate so a change in velocity is acceleration and so what that would look like if we have a velocity function well the output is feet per second the input is seconds and so we are figuring out what was the change in feet per second per second or how much did the velocity change in feet per second at that second so this is usually made a little bit more clear if I just look at a simple table now acceleration or deceleration is the change in velocity now a negative acceleration is what we would call a deceleration so let's let V of T be a velocity function in feet per second as a function of seconds so what this says right here is at time t equal one second we were traveling at a velocity of 10 feet per second so at this time we were traveling 10 feet per second per second what this says is at time two we were traveling 15 feet per second right that is how fast we were going at that moment now if I wanted to say well let's see I can answer this question we were going 10 feet per second and then we were going 15 percent 15 feet per second which means we sped up which means we accelerated so how much did we speed up well we went up five feet per second as we went over from one to two one second so our velocity increased five feet per second over that one second which is why we'd say five feet per second per second right the velocity increased five feet per second over a time of one second and so that's why this could make a little bit more sense we changed our velocity so many feet per second over a time interval of so many seconds so the slope or the derivative of velocity is acceleration if the units of velocity are feet per second the units of acceleration would be feet per second per second