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### Course: AP®︎/College Calculus BC>Unit 2

Lesson 1: Defining average and instantaneous rates of change at a point

# Derivative as a concept

Learn about derivatives as the instantaneous rate of change and the slope of the tangent line. This video introduces key concepts, including the difference between average and instantaneous rates of change, and how derivatives are central to differential calculus. Master various notations used to represent derivatives, such as Leibniz's, Lagrange's, and Newton's notations.

## Want to join the conversation?

• why slope of a line is not x/y?
(21 votes)
• There are two natural reasons as to why slope is Δ𝑦/Δ𝑥 instead of the reciprocal.
First, in everyday language, we say that something is steep if it has a large slope such that a small change horizontally corresponds to a drastic (large) change vertically. A mountain is steeper (has a greater slope) if for every 1 meter you go forward your altitude increases by 10 meters than if for every 1 meter you go forward your altitude increases by 2 meters. This everyday definition gives us Δ𝑦/Δ𝑥 for slope.

Also, in terms of a linear equation, 𝑥 is viewed as an independent variable, that is, the variable we have control over. We can set 𝑥 to anything we want. However, 𝑦 is the dependent variable. We have no control over its value – it is completely determined by 𝑥. So it is natural that we would want to gauge how much change in the dependent variable is caused by a unit change in the independent variable because we have control over the independent variable whereas the dependent variable is determined by the independent variable, not by us directly. This notion again gives us Δ𝑦/Δ𝑥 as the slope.
(125 votes)
• how could a point have a rate of change ? i mean it is a point a coordinate the change happen when we move from coordinate to other right ?
(31 votes)
• Hi Khalid,
In this case we are referring to instantaneous rate of change at the instant we 'get' to that point... the best way to visualize a rate of change at a point is to draw in a tangent line to the curve at that point... the slope of that line is your rate of change of the function at that point.

:)
(38 votes)
• what is the difference between `delta` x and dx?
(8 votes)
• Δx describes discrete change; i.e., you can say Δx = 1 or 0.1, and is probably used more in algebra.
dx represents an infinitesimal change, i.e., it doesn't have a value like dx = 0.0000001, but is simply infinitesimal (not a very rigorous explanation, I know). It's the calculus counterpart to Δx; because it's infinitesimal, a series of dx's put together can describe continuous change (as with derivatives and integrals).
(28 votes)
• How can a single point on a plot dictate a slope of a tangential line? There could be multiple combinations of y-intercepts and slopes with a single point. Can someone explain this to me more?
(9 votes)
• For any given point on a curve, there is only one line you can draw that will be tangent to that curve. As you go through and watch more videos, you'll find out how to take the derivative of an equation. When you plug x into that derivative equation, the result you'll get for y (or f(x)) will be your tangent line slope. Hope this helps!
(15 votes)
• Just out of curiosity, what happens if you take the derivative of a function's derivative? Is there a use for that?
(7 votes)
• Yes, that's called the second derivative. In fact, it's very useful for finding things like concavity (which way the graph curves). It'll be just a bit farther down the course.
(12 votes)
• I understand the concept explained in this video. A question arise now. Consider a graph between distance (in y-axis) and time (in x-axis). Now, if we take a derivative, what we do is that the change in the x value (dx) when dt is realy close to zero (infinitely small). Usually, dx/dt is known as the velocity. Thats Okay. But, how the unit is m/s (meter-per-second) even though we use infinitesimally small time?
(10 votes)
• This is informal but let's say the distance changes twice as fast as the time. The ratio is always 2:1, no matter how big or small. The small time is also cancelled out by the small distance.
(7 votes)
• so just checking my understanding, in this video, the slope of a secant line is used to calculate the average rate of change and the slope of a tangent line is used to calculate the derivative?

And any relation between the secant with the curve here and the secant we learned in trigonometry?
(8 votes)
• Yep! You can also say that the slope of the tangent is used to get the instantaneous rate of change at a point. The "at a point" is important as the secant gives the rate of change "between two points"

There actually is! Check this (https://en.wikipedia.org/wiki/File:Unitcircledefs.svg) and you'll see why the line is called the secant line. You can also see why the tangent line is called so.
(9 votes)
• When he says the change in x getting ever closer to 0, is he referring to the fact that as we approach our desired point, there is less and less change in the x value?
(8 votes)
• Essentially, yes. The less change in the x-value, the more accurate the slope is at the desired point.
(7 votes)
• Consider the graph of y=|x|.
What would the derivative be at x=0? I'm wondering this because it intuitively feels like there should be infinite possible tangent lines to that point. Can a point have more than one derivative?
(6 votes)
• Nice question!
You are right that in a sense, this derivative is ambiguous. The derivative of |x| at x=0 does not exist because, in a sense, the graph of y=|x| has a sharp corner at x=0.
More precisely, the limit definition of this derivative is

lim h-->0 of (|0+h|-|0|)/h = lim h-->0 of |h|/h.

Since lim h-->0^+ of |h|/h = lim h-->0^+ of h/h = 1, but
lim h-->0^- of |h|/h = lim h-->0^- of -h/h = -1, we see that
lim h-->0 of |h|/h does not exist.

So this derivative does not exist! Note that this example shows that it's possible for a function to be continuous at a point without being differentiable there.
(11 votes)
• At , Sal draws a tangent line that would tell you the derivative(I think), but can't you draw another line with a slightly different slope and have it touch just that specific point on the original graph? Can't you have more than one tangent line?
(4 votes)
• You can't. For any point on a curve, you can have a single tangent line. If you were to slightly change the slope, the "tangent" would eventually hit the curve twice, making it a secant instead.
(12 votes)

## Video transcript

- [Instructor] 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 or 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 would be the slope of this line of the secant line, but if we picked 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 x one. 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 these sprinters, Usain Bolt example. If we wanted to figure out the speed of Usain Bolt at a given instant, well maybe this describes his position with respect to time if y was position and x is time. Usually, you would see t as time, but let's say x is time, so then, if were talking about right at this time, we're talking about the instantaneous rate, and this idea is the 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 an exclamation mark because it's so conceptually important here. So how can we denote a derivative? One way is known as Leibniz's notation, and Leibniz 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 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, say that point and that point, and then let's get even closer and that point and that point, and then let's get even closer, and let's see what happens as the change in x approaches zero, and so using these d's instead of deltas, this was Leibniz's 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's 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 will 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 x one. So this notation takes a little bit of time getting used to, the Lagrange notation. 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 is y with a dot over it, which also denotes the derivative. 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 could imagine, 'cause 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 a point. We're going to be able to figure out general equations that described the derivative for any given point, so be very, very excited.