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Differentiation Method

by Batool Akmal
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    00:00 So the next question that we're going to answer is what does differentiation actually do.

    00:06 Why are we differentiating functions and why are we using these complicated and easy methods to find derivatives.

    00:13 Differentiation helps you to find the gradient of a curve at a particular point.

    00:18 You may think that's a bit strange because the gradient of a curve changes all the time.

    00:22 But you can find the gradient of a particular point on the curve just by substituting your x and y values into that point.

    00:31 It really is quite amazing how that works.

    00:33 That when you differentiate, you get a general gradient of the curve.

    00:37 And then you can look at any specific point by just substituting numbers into it.

    00:41 We'll try this out with some examples shortly.

    00:44 First of all we'll look at practicing A, differential question.

    00:50 So we can do this really quickly now because I won't make you do this using first principles.

    00:54 We're just going to make sure that we can differentiate and differentiate correctly.

    00:59 And then we'll move on to substituting numbers in finding the gradient at more specific points.

    01:05 And eventually we'll move on to finding gradients of tangents and lines and even equations of tangents of lines at those specific points.

    01:12 So let's have a look at this question.

    01:14 We will differentiate y = 3x2 + 5x - 1 and we'll do it the faster way.

    01:20 So remember what we said that dy by dx.

    01:24 You bring the power to the front.

    01:27 So we'll have 2.

    01:28 3x, 3 stays as it is.

    01:32 Decrease the power of x by 1.

    01:34 If I write it out as the whole thing.

    01:36 You still bring the power of x down.

    01:39 So the power of x now is 1.

    01:41 And then you have 5x to the 1 - 1 which just becomes x to the 0.

    01:47 And then the constant in the end will disappear because it has a power of x to the 0.

    01:52 This than gives you an answer of 6x + 5.

    01:57 and that's your gradient.

    01:59 So really that's now made our lives a lot easier because we don't have to apply differentiation the first principles, we can just do it the faster way.

    02:06 Remember now that this isn't just dy by dx.

    02:10 This is actually telling us something.

    02:13 It's telling us the gradient of this equation.

    02:15 It's telling us how steep it is and at any different, any given point.

    02:20 So say for example if this is a curve.

    02:26 And the equation to that curve is y = to 3x2 + 5x - 1.

    02:32 If we were to sketch this curve, you could factorize this.

    02:38 So we've got 3x and x.

    02:41 And then you'll have 1 and 1.

    02:43 If you try this out, you will see that no matter what signs that you put in here and you won't quite be able to factorize it.

    02:50 So they are other methods if you recall.

    02:53 So previously if you remember that you would have done the quadratic formula which states the x = -b+-squared(b-4ac)/2a.

    03:07 Your a is the number next to the x squared.

    03:12 So this is your a.

    03:13 Where b is the number next to the x and c is the number by itself.

    03:17 So if ever you are not able to factorize to find points of intersections of a curve, remember there's always backup plans that we could use this equation to find our solutions.

    03:29 So in this case if we just do x = b is 5 so we're going to have -5, +- 5(squared) - 4, a is 3 and c is -1.

    03:44 All over 2a.

    03:49 You'll see we will end up with -5 plus or minus.

    03:52 Here we have 25 - 12 over 6.

    03:59 We are now going to end up with 2 two solutions.

    04:02 So if I just write out here.

    04:05 Let's just see if we can squeeze that in.

    04:07 So that gives me root 13 over 6.

    04:10 So you'll have either a minus 5 plus root 13 over 6.

    04:15 Or minus 5, minus root 13 over 6.

    04:18 We don't have calculators here.

    04:20 So we're just going to roughly say that one of them is a positive answer and one of them is a negative answer.

    04:25 So if I sketch this, just giving over a rough idea.

    04:31 So we got a positive and a negative answer.

    04:33 And we've got a quadratic.

    04:37 What we just found for ourselves is that the gradient of this curve at any given point, so here or here or here, is going to be 6x + 5.

    04:46 And all you have to do is to find the actual gradient at that point is to substitute this x value into this equation.

    04:54 So you could find the gradient at this point.

    04:56 If you know this x value, you're just substituting here, or you could find the gradient at this point.

    05:02 You know the x value you just substitute in here to find your specific gradient at that point.

    05:07 Let's look at our second example now.

    05:11 Now this is the only way that they can really make our lives difficult, is by giving us differential or equations to differentiate which are little bit more complicated.

    05:20 So you have square roots or you have x values at the bottom as denominators.

    05:25 And how do we tweak it and how do we make it more differentiation friendly for ourselves.

    05:30 So let's look at this question and see if we can make it a little bit easier.

    05:34 And just apply the straight forward concepts of differentiation.

    05:37 So I'm just going to write this question again so that we can move things around.

    05:41 So we've got y = 4 over x cubed, plus 2x to the power of 5 minus root x.

    05:48 Now the first thing that we need to do in order to make our lives little bit easier, is we don't want this as denominator.

    05:55 So we need to bring this up.

    05:57 And secondly we don't want things like these.

    05:59 So we don't want roots.

    06:00 We want powers.

    06:01 Because powers we can deal with and we know what to do with the power of something.

    06:06 We can bring it down and we can decrease that by 1.

    06:09 So before I differentiate I'm going to rewrite this.

    06:12 As you know by the rule of indices or the rule of denominators being at the bottom, you are allowed to bring this up.

    06:20 So you can bring this value up, as long as you change the sign of the power.

    06:24 So we can rewrite this as 4x to the minus 3.

    06:28 So this is one of the rules of the indices.

    06:30 So we've got 2x to the power 5 which doesn't bother us.

    06:33 And then we've got x to the power of a half.

    06:36 So remember now I haven't differentiated it.

    06:39 I've just rewritten it in order to make it easier for myself.

    06:42 And this is perhaps the only way that these differentiation questions can become a little bite mathematically challenging.

    06:48 But all you have to do is just look at it, simplify a little bit for yourselves.

    06:53 Make it more differentiation friendly and then we just apply the same rule over and over again no matter what they are asking.

    07:01 So here if I dy by dx and then I'm going to bring the power down so minus 3.

    07:07 I've got 4x to the minus 3 of minus 1.

    07:11 So decrease the power by 1.

    07:12 You can bring 5 now.

    07:14 The 2 is a constant which stays as it is.

    07:17 And then you take 1 away from it.

    07:20 And now we bring the half down and remember it doesn't matter what the power is.

    07:23 No matter how complicated that is.

    07:25 You just apply the same rule to it.

    07:27 And then I've got x to the half minus 1.

    07:30 And then I've basically done the difficult thing.

    07:34 All I have to do is tidy it up and that will give me my final answer.

    07:38 So this gives me -12 when I multiply these two numbers together.

    07:42 I have x to the minus 4.

    07:45 5 times 2 gives me 10.

    07:46 x to the 4.

    07:47 And then I have minus the half x to the minus the half.

    07:53 And if you ever struggle with working something like this out, you can always just do it on the side because you are not quite using a calculator a half minus 1.

    08:01 You want 2 to be your common denominator.

    08:04 So you want both of these numbers to be above 2.

    08:06 This changes, this doesn't change any.

    08:09 This stays as it is.

    08:10 And in order to increase this whole fraction by 2, you multiply the bottom, with the top by 2 as well.

    08:17 So that gives you minus 2 to give you minus 1 at the top and 2 at the bottom.

    08:22 So that's where it comes from.

    08:23 So if fractions ever become difficult and you're not using a calculator, there's no shame in working out on the side and then putting it back into your answer.

    08:31 Final few things that we could do so, we can make this x to the minus 4 go down.

    08:37 So we can make it positive, so that becomes x to the 4.

    08:39 Plus 10x to the 4.

    08:41 And then we can bring this negative x to the minus the half down.

    08:46 So that becomes joins the 2 at the bottom.

    08:49 So that becomes the half.

    08:50 And if you rewrite this as a square root, we can also do that.

    08:54 And all you can do now is just change this last setup here if you wanted too.

    08:58 So you could write this as square root of x.

    09:01 Just the way that they did in the question.

    09:03 So we have 12 over x to the 4 plus 10x to the 4 and then -1 over 2.

    09:12 And instead of x to the half, you can write as root x just to show that you understand where it comes from.

    09:18 So once again you are using a rule of indices to convert it back and make it look tidier.

    09:23 So the gradient of this complicated function here is given by this.

    09:29 I'm not going to try and sketch this out.

    09:32 But remember that you can't find the gradient at any particular point on this curve just by making substitutions for these x values here.

    09:42 So if for example you are looking at the gradient at x equals to 2.

    09:46 All you have to do is substitute 2 instead of the x's and you will get your gradient numerical value, be it positive or negative.

    09:53 It can tell you a lot about what the curve is doing.


    About the Lecture

    The lecture Differentiation Method by Batool Akmal is from the course Calculus Methods: Differentiation.


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    Author of lecture Differentiation Method

     Batool Akmal

    Batool Akmal


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    best for a beginner...
    By sheikh d. on 06. February 2017 for Differentiation Method

    simple presentation with examples..learning mathematics makes economical in typing words