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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)
Introduction to the idea of a derivative as instantaneous rate of change or the slope of the tangent line.

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  • blobby green style avatar for user Zahoor Islam
    why slope of a line is not x/y?
    (18 votes)
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  • aqualine ultimate style avatar for user khalid
    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 ?
    (2 votes)
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    • duskpin ultimate style avatar for user Ms Demaray
      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.

      :)
      (27 votes)
  • leaf green style avatar for user Joy
    what is the difference between delta x and dx?
    (4 votes)
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    • leaf grey style avatar for user Alex
      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).
      (14 votes)
  • aqualine ultimate style avatar for user lailasyed1a
    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?
    (7 votes)
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    • orange juice squid orange style avatar for user Silas Thomas
      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!
      (7 votes)
  • leaf green style avatar for user sst
    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?
    (5 votes)
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  • aqualine ultimate style avatar for user Insatiable
    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?
    (4 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      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.
      (6 votes)
  • starky sapling style avatar for user 20leunge
    Is the secant he is referring to the one defined as a line that intersects 2 points on a curve?
    (4 votes)
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  • blobby green style avatar for user Dylan
    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?
    (2 votes)
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  • primosaur ultimate style avatar for user Steve
    Hey everyone...so I have a question that may be more 'semantics'. But when I am getting more into physics I see a lot of talk about "Deriving an equation"...or something like "Derive the kinematic equations" etc... Is this the same meaning as literally taking the derivative of the equation? Or does "deriving" in this sense mean something else?
    (3 votes)
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    • primosaur ultimate style avatar for user Yellow Shiƒt»
      They are two different meanings.

      Deriving an equation in physics means to find where an equation comes from. It is somewhat like writing a mathematical proof (though not as rigorous).

      In calculus, "deriving," or taking the derivative, means to find the "slope" of a given function. I put slope in quotes because it usually to the slope of a line. Derivatives, on the other hand, are a measure of the rate of change, but they apply to almost any function. Think of them as an extension of the concept of slope.
      (3 votes)
  • purple pi purple style avatar for user The first integral proponent
    What does the dx in the fundamental theorem of calculus means?
    (3 votes)
    Default Khan Academy avatar avatar for user

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.