Differentiation

hello and welcome to starfish Maps my name is Sarah and today I want to look at differentiation this is a great topic I would really enjoy differentiating and I hope you will too I'm going to start by showing you how to differentiate with six examples increasing difficulty now more look at for example questions with a bit more problem-solving to them they get a little bit tricky by the end so whether you're a complete beginner to differentiation or you're just home quick brush up there should be something in here for you as I go grab a pen and paper pause the video and have a go as well if you feel you can I hope this is helpful let's get started okay so here's a fairly basic polynomial equation and to differentiate this I want you to remember two times and take if you can always remember to time some take and we'll help you when you do more complex differentiation and especially when you do integrations because when people start learning to integrate they can get the two things mixed up quite easily so remember two times and take I'll show you what I mean we're going to times of the three down to the 5 so 3 times 5 is 15 then we're going to take so that 3 we take 1 you always take one away so it's going to become 2 next term times and take one away so it'll just be one in fact then we don't even need to write that one next we've got 8x now here it doesn't have a power that you can think about if you like is having power of 1 so x and take when you take one away to be 0 and X the power of 0 is 1 anything suppose there is 1 so that term we can just get rid of that it will just be 8 so in general if you've got a number of times the less only differentiating you just leave that letter and you get the number left if we differentiate 3 similarly it doesn't even have a neck so that will completely vanish and that's all we're left with so every time you're kind of losing a power of X so now we've different to show that we differentiated the language we use is actually d-y-by-dx and that's just saying we differentiated Y with respect to X so that X from the bottom is saying that's the variable we can use that ok I'm gonna give you six examples now and they are going to be getting more complicated quite quickly actually and so have you get at the ones that you feel you can and we'll go through them okay so do pause the video here if you want to have a go at some of these yourself otherwise we can do this together the first one we're going to time some take and the X will vanish and so will one and now on this one I've actually used slightly different language of these f of X sometimes they use Y sometimes use f of X it's good to be exposed to both those languages if you've differentiated f of X the way to show differentiate it is to do it in a - just F dash of X and that's all that means the same as dy by DX it's just a different way of writing it now all of these other ones are getting harder you need to prepare them first before you can differentiate what I mean by that is you need to write them in a way that they've all got powers so here that 4 over X isn't very helpful at the moment we need to write that as a power of X so if you need to brush up on your indices work do watch my other video otherwise you should know about X to the power of minus 1 now we can differentiate x and take x the most 1/4 is months for when you take one away from minus one you actually get minus to watch out for that some people might be tempted to say it's zero but it's not ok let's carry on this one needs to be written as powers as well okay Weldon if you got that one right just check you've got your powers I care in the fraction work is all right let's carry on to prepare this while we need to expand the bracket now I can differentiate no problem okay these last two have got fractions in these are quite typical sorts of questions to get you need to prepare them to differentiate of course and there's a couple of ways of doing this and my favorite where at the moment is probably to bring that up as a power of minus one and then expand the brackets outs remember when you expand brackets that when you times the terms you actually add the powers so that first one will be three and minus one which is now we can differentiate okay well done if you got that on right so far let's differentiate brilliant well done if you got that one right there's a few mistakes you could make there potentially these fractions aren't particularly easy so make sure your fraction work is really secure well done that's just a handful of questions but I do recommend that you keep practicing those and do some from a textbook or from wherever you get your questions from differentiating needs to be right on your fingertips you need to be able to do it really easily without even much thinking so keep practicing until we get to that point all right I'm gonna give you some exams start questions now so far I've shown you how to differentiate but I haven't told you why we differentiate it's actually a really powerful tool and its really really useful in months the biggest reason to differentiate is it gives you the gradient of a graph so if there's only one thing you take away from this video today remember differentiating gives you the gradient so far you've been able to find the grid into a straight line where the gradients are same all along it but for anything that's not straight for any curve the gradients constantly changing so to be able to find a gradient at any point you need to differentiate and then put in the x value for where you want to find the gradient obviously it'll be different for every point on the graph so differentiating gives you gradients if we're asked now to find the gradient where X is 8 we need to differentiate to be able to differentiate this we need to prepare it so let's write that as a power of X and now I can differentiate we're asked for the gradient where X was 8 so we need to substitute 8 back into this but to be able to substitute at the moment it's not a very user-friendly version so let's rewrite that with the power back down on the bottom and we can go one step further and write that X to the power of 1/3 as the third root of x okay now it's written like that we can use it a little bit easier so let's substitute 8 in the third root of 8 is 2 so so when X is 8 the gradient is 4/3 again this is the question about gradient so we know we're going to differentiate this time were asked to find where the gradient is 0 so let's start by differentiating and then we'll put that equal to 0 we know the gradient 0 so we can replace that and now we've got a nice easy quadratic to solve if you write down this question you might notice that it actually says find the coordinates not just the x coordinates so we need to find the Y coordinates as well so we need to put them back into the original equation to get Y just be mindful of those negative signs there and meinster squared is positive and then you're taking it away so it's still negative well done if you got that one right I haven't figured out how to write math these stuff when I type text in here so I've actually sped the second derivative and worse but what that normally means what you'll normally see is d squared y by DX squared and the second derivative just means that you differentiate twice so you can actually get the second derivative third derivative fourth derivative and so on just by differentiating it again and again so we're just going to differentiate this and then do it once more to find the second derivative before we differentiate it we need to prepare it the bus the second derivative this is quite a wordy question so it might be worth taking a screenshot or making a note of the question so pause the video if you need to it looks quite complex but if we just take each piece of information at a time and set up an equation we'll be okay so the first piece of information we have is that it passes through 0 minus 4 that's just an X and the y coordinate they've given us so let's substitute those back in those times will cancel because they're zeros so we're just left with C is minus 4 which is nice and easy the second piece of information is about the gradient so it makes sense differentiate this when I differentiate that I just treat the a and B as numbers so I almost ignore them bring the 2 down to the front times and take the same as I would with numbers okay we're told that the gradient is minus 2 when X is minus 1/2 there's not a lot more we can do to that right now so let's move on to the third piece of information and see if it will help us to come back we need the second derivative so that's differentiate again that's got rid of some more stuff so let's substitute in what we know we know it's 10 when X is minus 1/2 again actually this doesn't even have an excellent so it'll just equal to a which means that a is 5 now we can go back to here and substitute a back in and we've done it very well done I hope that's helped you in some way keep practicing those and thanks for watching have fun

Channel: Starfish Maths

Views: 999,181

Rating: 4.861958 out of 5

Keywords: AS, Maths, MEI, OCR, level, Core, C1, differentiation, differentiating, derivative, gradient, gradient of a curve, finding, second derivative, edexcel, AQA, Core 1, Core 2, AS Maths, AS calculus, calculus, dy by dx, functions, AS Level, A Level, GCSE

Id: BcOPKQAZcn0

Length: 11min 27sec (687 seconds)

Published: Tue Oct 25 2016