If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Derivative as a concept

AP.CALC:
CHA‑2 (EU)
,
CHA‑2.A (LO)
,
CHA‑2.A.1 (EK)
,
CHA‑2.B (LO)
,
CHA‑2.B.1 (EK)

## Video transcript

you are likely already familiar with the idea of a slope of a line if you're not I encourage you to review it on Khan Academy but all it is it's describing the rate of change of a vertical variable with respect to a horizontal variable so for example here I have our classic y-axis in the vertical direction and x-axis in the horizontal direction and if I wanted to figure out the slope of this line I could pick two points say that point and that point I could say okay from this point to this point what is my change in X well my change in X would be this distance right over here change in X the Greek letter Delta this triangle here it's just shorthand for change so change in X and I could also calculate the change in Y so this point going up to that point our change in Y would be this right over here our change in Y and then we would define slope or we have defined slope as change in Y over change in X so slope is equal to the rate of change of our vertical variable over the rate of change of our horizontal variable sometimes described as rise over run and for any line it's associated with a slope because it has a constant rate of change if you took any two points on this line no matter how far apart are no matter how close together anywhere they sit on the line if you were to do this calculation you would get the same slope that's what makes it a line but what's fascinating about calculus is we're going to build the tools so that we can think about the rate of change not just of a line which we've called slope in the past we can think about the rate of change the instantaneous rate of change of a curve of something whose rate of change is possibly constantly changing so for example here's a curve where the rate of change of Y with respect to X is constantly changing even if we wanted to use our traditional tools if we said okay we can calculate the average rate of change let's say between this point and this point well what would it be well the average rate of change between this point and this point would be the slope of the line that connects them so it'd be the slope of this line of the secant line but if we pick two different points we pick this point and this point the average rate of change between those points all of a sudden looks quite different it looks like it has a higher slope so even when we take the slopes between two points on the line the secant lines you can see that those slopes are changing but what if we wanted to ask ourselves an even more interesting question what is the instantaneous rate of change at a point so for example how fast is Y changing with respect to X exactly at that point exactly when X is equal to that value let's call it x1 well one way you could think about it is what if we could draw a tangent line to this point a line that just touches the graph right over there and we can calculate the slope of that line well that should be the rate of change at that point the instantaneous rate of change so in this case the tangent line might look something like that if we know the slope of this well then we could say that that's the instantaneous rate of change at that point why do I say instantaneous rate of change well think about the video on the sprinters it was saying bolt example if we wanted to figure out the speed of Usain Bolt at a given instant well maybe just describes his position with respect to time if Y was position and X is time usually you would see T is time but let's say X is time so then if we're talking about right at this time we're talking about the instantaneous rate and this idea is a central idea of differential calculus and it's known as a derivative the slope of the tangent line which you could also view as the instantaneous rate of change I'm putting exclamation mark because it's so conceptually important here so how can we denote a derivative one way is known as live missus notation and liveness is one of the fathers of calculus along with Isaac Newton and his notation you would denote the slope of the tangent line as equaling dy over DX now why do I like this notation because it really comes from this idea of a slope which is change in Y over change in X as you'll see in future with videos one way to think about the slope of the tangent line is well let's calculate the slope of secant lines let's say between that point and that point but then let's get even closer and say that point in that point and in let's get to even closer than that point in that point then let's get even closer and let's see what happens as the change in X approaches 0 and so using these DS instead of deltas this was liveness his way of saying hey what happens if my changes in say X become close to zero so this idea this is known as sometimes differential notation Leibniz notation is instead of just change in Y over change in X super small changes in Y for a super small change in X especially as the change in X approaches zero and as you'll see that is how we will calculate the derivative now there's other notations if this curve is described as y is equal to f of X the slope of the tangent line at that point could be denoted as equaling f prime of x1 so this notation it takes a little bit of time getting used to the Lagrangian it's saying F prime is representing the derivative it's telling us the slope of the tangent line for a given point so if you input an X into this function into F you're getting the corresponding Y value if you input an X into F prime you're getting the slope of the tangent line at that point now another notation that you'll see less likely in a calculus class but you might see in a physics class is the notation Y with a dot over it so you could write this as Y with a dot over it which also denotes the you might also see y-prime this would be more common in a math class now as we march forward in our calculus adventure we will build the tools to actually calculate these things and if you're already familiar with limits they will be very useful as you can imagine because we're really going to be taking the limit of our change in Y over change in X as our change in X approaches zero and we're not just going to be able to figure it out for point we're going to be able to figure out general equations that describe the derivative for any given point so be very very excited
AP® is a registered trademark of the College Board, which has not reviewed this resource.