Gradients of Straight and Curved Lines

by Batool Akmal

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    00:01 So let's look at gradients of straight lines and curves.

    00:04 Firstly, we'll start with a little recap.

    00:06 We should all know at this point that to measure the gradient of a straight line, you need to take the change y over the change in x.

    00:16 You're measuring the steepness or the gradient of any type of surface that is tilted.

    00:21 So you're looking at how the y axis changes divided by how the x axis changes.

    00:28 We sometimes call this gradient m and we use the notation delta y over delta x or changing y over changing x very simply.

    00:36 Now let's move on to imagining that we are doing this to a curve.

    00:42 So let's take that idea now and develop our curve, here's my x-axis and my y-axis.

    00:52 So imagine that we have a curve, and now we start to look at this curve and we start to look at how we'll find the gradient of this curve.

    01:01 Now imagine the ancients did something like this, they took a point here, x which gave you an answer of f of x, so let's call that the function of x.

    01:11 And if you change this x value by a small amount, let's call that delta x.

    01:16 We get to a new point which we will now call x plus delta x.

    01:20 As a result, this will give you a new value on your y-axis, and let's call this f of x plus delta x.

    01:28 So it's almost like we're looking at this lying in between, so it's a straight line and we're looking at the gradient of this straight line here.

    01:37 How do we do this? So we use the same concept as we did previously using a straight line.

    01:43 We said that the gradient which we'll call m is the same as delta y over delta x.

    01:50 And we now know that this is the change in y, so change in y and this here is the change in x.

    01:57 and you can actually see the change in y and the change in x here, on your picture.

    02:04 So here we have the change in y, and here we have the change in x.

    02:09 So let's work out what the change in y would be here, the change in y would in this case be f of x plus delta x and you subtract from it, f of x, to get this value here that we've just worked out.

    02:24 And in similarly for the x-axis, we'll have x plus delta x and we take away x from it.

    02:32 Let's put that into our formula now, so, we're going to say that delta y over delta x is following from what we said earlier, is the change in y over change in x.

    02:43 So our change in y over change in x is this. So we have f of x plus delta x minus f of x and we divide that by our change in x, which is this part here.

    02:56 So that's x plus delta x minus x.

    02:59 You can see now that we can simplify the bottom denominator a little bit.

    03:05 So we've got plus x here and minus x, so that cancels out.

    03:09 We can make this a little bit fancier by using a different notation for delta y over delta x.

    03:16 We can write this as dy/dx, which is basically the notation that we use for differentiation and if we rewrite this, we get f of x plus delta x minus f of x, all over delta x.

    03:29 Now this here is the definition of a derivative, but we also call it differentiation from first principle.

    03:40 So it's differentiation done by the first principles.

    03:45 You will see that it's used in different types of questions, especially questions that are a lot more difficult to differentiate.

    03:55 So you can always just come straight to the basic definition of differentiation.

    03:59 However, there's one little problem here, like I said before that we were looking at the gradient of this line and differentiation is basically finding the gradient of a single point.

    04:13 So what we need to do is we need to reduce the length of this line.

    04:17 So what we're trying to do is to make delta x smaller and smaller and smaller and essentially that means that we want delta x to tend to zero.

    04:27 We want to make this change so small, that this comes here and you were essentially then finding the gradient of one singular point there.

    04:36 So, if we now apply that concept here, if we say dy/dx as the limit of delta x tends to zero is the same, so we're now saying the f of x plus delta x minus f of x over delta x.

    04:55 And this is your formal definition of differentiation from first principles or like I said before, it's known as the definition of a derivative.

    05:08 You will see that we will use this definition lots of times through this course, especially when we're faced with functions that aren't as simple to differentiate.

    05:16 We will always go back to the basics or to the definition of a derivative.

    About the Lecture

    The lecture Gradients of Straight and Curved Lines by Batool Akmal is from the course Calculus Methods: Gradients and First Principles.

    Included Quiz Questions

    1. Δy / Δx
    2. Δx / Δy
    3. Δy * Δx
    4. Arctan(Δy/Δx)
    5. Δx + Δy

    Author of lecture Gradients of Straight and Curved Lines

     Batool Akmal

    Batool Akmal

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    a very good explanation
    By Anna B. on 06. March 2017 for Gradients of Straight and Curved Lines

    the explanation was very clear , i've understood it very quickly , the vidéo is realy well done .