A LEVEL MATHS: Differentiation First Principles Exam Practice! LESSON 11

hey guys just to let you know that you can catch these lessons live every Thursday from 6PM until 7 PM UK or BST time and I'll just allow you to ask any questions that you want and get some answers to them without further Ado let's get on with the lesson hey guys welcome back to another a-level tuition video today we're going to be covering differentiation and differentiation from first principles so first of all what is differentiation well it's a way of getting a function to tell you the gradient at any value of x you might think well why do I need this right well the normal way of finding well the original way of finding the gradient of a function is if it's a straight line all you do is you find the difference of two points right and it works perfectly works perfectly fine however most functions are not straight lines right instead we tend to have things like exponentials or we could have a kind of a cubic graph or a quartic graph so most graphs are all Wiggly now if I want to find the gradient here I can use maybe the difference you know this kind of method here both want to find let's say on this curved part I have to draw a tangent and find the gradient of the tangent and then what if I wanted to change it and find this one but then again I have to draw a tangent and the problem with this is one drawing the tangents is always an estimate because you can always angle it slightly differently and two it's tedious right if I want to like move slightly over to the right then all of a sudden I need to redraw a whole another tangent so this is not a good method for us to use so instead it's very helpful to use differentiation as opposed to that and that will create a function that will tell us what the gradient is at any point of the gradients constant will just give us a number so the question is how do we get this well it actually uses mathematically speaking it uses the same Principle as drawing a tangent however the tangent is a going to be perfect because it's done mathematically rather than graphically and B we make it vanishingly small so we get an equation for what the gradient will look like at any given point so that's all well and good so we need to discuss how we're going to do this well the first thing we're going to do is we're going to work it out from first principles so this is essentially how do we do this mathematically from scratch okay now I would like to um preface this differentiation comes back in A2 maths and it's a lot more complicated there's a lot more functions let me differentiate so sine cos and tan exponentials logarithms Etc and the thing is we don't have to do it from first principles we have predefined results okay so for example the differentiation of sine X will always give you cos x okay you don't have to prove it but for the sake of this so you can see what's happening with differentiation we need to be able to do first principles and there are questions that literally say show that the differentiation of this function is this from the first principle so we need to know how to do it but again when we're just normally differentiating we don't need to prove it this way so what we do is we will have some kind of function f of x and to differentiate we normally just put a little apostrophe that means it's the differentiated version and the way we take this and you're not going to have seen this before but I promise you it's not that bad now those of you that are actually interested in maths and doing math to kind of University or physics and things like this this might look a bit exciting to you slightly but essentially what we're going to do is we're going to take our function and add h okay then we're going to subtract the original function divided by H and then see what happens when we reduce H to zero we're not going to make it zero we're going to gradually reduce it and see how the function behaves and that is how you find differentials okay so that might seem a bit a bit far-fetched but if I give you some examples I can show you how it's done okay so so prove from first principles that the derivative of f of 4X is four so what we're going to do is use our little formula up above okay so our function that we're trying to differentiate and again when you differentiate you find the derivative okay so the derivative of 4X is 4 means if I differentiate 4X I should get the number 4. okay so so if I do this f of x is equal to 4X now f of x plus 4 I'm replacing x with X plus 4 so that would be 4 X Plus 4. okay H sorry now if I look at this formula here let's substitute it in shall we so F Prime X so the first derivative is limit as H tends to zero of f of x plus h which is four X Plus H minus the original function which was 4X over h so again we don't do this now because obviously it's h tends to zero it looks like the function is going to go off to Infinity because we're dividing by something that's approaching zero but let's actually have a look at what happens so we expand the top as usual so we have 4X plus 4H minus 4X over h then all we need to do is simplify this so 4X minus X is going to give me 0 and then we have four h over h okay which again will just equal four and again this is for the limit of H tends to zero so again this thing looks a bit fancy but for the purposes of as maths it's not really that bad okay all it's imagining is imagine if H is zero so if I imagine H is zero in this case can you see that that would just give me four because there is no h so H is just going to equal zero so it's just going to equal four so we've proved from first principles that the derivative of 4X is four so you don't have any questions about that [Music] so again it is a bit obscure and the next one that we're doing is actually going to be a bit harder it's a bit more complicated hopefully that's okay perfect so from first principles that the derivative of x cubed is 3x squared so again let's define what our function is we're trying to differentiate X cubed so f of x plus h again all we do is we replace x with X Plus H so we have X Plus H cubed then our derivative is the limit again I'm going to be a bit lazy and hopefully I'm not going to write limit every single time until I actually take the limit and we have f of x plus h so we have X plus h cubed again I'm just using the woman here minus f of x so minus X cubed over h okay so if we write what that equals to we have to now expand the brackets unfortunately so if we expand the first two brackets we're going to have x squared right Plus 2hx plus h squared and that's still going to multiply by X plus h minus x cubed so now we have X cubed plus 2 h x squared plus h squared x Plus h x squared Plus 2 H squared x plus h cubed minus x cubed over h so we have a lot here so what we're going to do is nice and quickly cancel this down so X cubed cancels with the X cubed now how much two h x squared do we have well we have two h x squared we also have another HX squared so that gives me 3 HX squared so that covers that then we have three H squared X and then lastly H cubed over h and what I'm going to do is just divide through by h so cancel cancel so I'm left with the limit as H tends to zero of 3x squared Plus 3hx plus h squared okay so if we ever think about that if H is becoming zero these two terms will go to zero right as I make H smaller those terms get smaller and smaller and smaller and smaller and smaller so the only thing I have left is 3x squared so again F Prime of x would just be 3 x squared so again proof and first principles are the derivative of 2x cubed is six x squared so we have f of x equals 2 x cubed so then f of x plus h will be 2 X plus h or cubed so again if we substitute into our formula that means F Prime of x equals the limit as H tends to nothing of f of x plus h sub 2 X plus h cubed minus the original 2x cubed over H we can kind of cheat because we just did X Plus H cubed right but let's be a bit authentic and let's actually do it again so we have two x squared Plus 2hx that's H squared times X plus h minus 2x cubed over H so that means we have to x cubed Plus 2 h x squared Plus H squared x plus HX squared plus 2 H squared X Plus H cubed minus two x cubed over h again we're going to other like terms we have X cubed we have so we can count that one out we have two HX squared and HX squared so that'd be plus 3 h then we have three h squared x so we can get rid of those two and then H cubed minus then we can finally expand the brackets and we have so we can actually cancel the 2x cubed and the 2x cubed here so that leaves us with 6 h x squared Plus whoops plus 6 h squared X plus two H cubed over h which we can now cancel a h from everything and we're left with the limit again I am being a bit lazy and not writing limit every single time but that would make me want to kill myself plus HX plus 2 H squared and once again if we have a look if I make this uh H approaches zero then these two terms are going to disappear which leaves me with 6X squared which again is what the answer is so today we'll be doing all of the differentiation from first principles we'll do a proper differentiation next week those questions can be a bit tricky but again this is just to show you what you need to do and again please keep in mind that in A2 there are some first principle proofs that are a bit trickier we like sine and cos for some exam boards so getting really good at this is pretty damn vital even though it may seem very tedious for you at the moment squared which means far f of x plus h will be five X plus h squared so prove that is 10 H so once again F Prime X is the limit H tends to zero of f of x plus h so we have five X plus h squared minus five x squared over h so again we can start pumping this out so that gives me 5 x squared plus 2hx plus h squared minus 5x squared H once again this 5x squared is going to count with this 5x squared so we're just left with 5 times 2hx would be 10 HX pause 5 H squared over H because we can cancel out H from everything and again you should find that in most in almost all cases H is going to cancel which should hopefully make sense because what would happen if this h on the bottom doesn't cancel well depending on how depending how well versed you want maths to just an a-level brain when H turns to zero it means our fraction is going to approach Infinity because we're dividing by a number that's smaller and smaller and smaller again for those of you that may know a bit more maths there are some other weird things that can happen but in this case which has F Prime of X is going to now equal to the Limit as H tends to zero of ten X plus 5H so again if you have a look at the if I imagine that this is becoming zero this term here is going to get smaller and smaller and smaller while this term isn't going to change at all which means as H approaches 0 we can say that this thing goes away so that means F Prime of x is simply equal to 10 x which once again is what they've asked me to show this will be the last question we did today again this was a fairly short lesson but it's not too bad we're going to do the proper differentiation next week gets a bit more tricky gets a bit more fun two from first principles that the derivative of k x cubed is three KX squared and we'll actually finish this video off with me kind of showing you what the um shorter way to do it is okay but again you do need to know first principles I'm afraid so we have f of x is KX cubed and we also have f of x plus h oops is K X plus h cubed so once again [Music] of K X plus h cubed minus KX over so that equals uh once again we're gonna have to do this fun business we have x squared plus 2hx plus h squared and there's still an X plus h on the end of that minus k x cubed and we can start expanded brackets again we have X cubed Plus 2 h x squared plus h squared x right so just for as you know I'm doing the timesing Everything by X first then times everything by h close the brackets and all of that's over h so we have K and now we can actually start simplifying so we have X cubed that's fine so we can cross that one off two h x squared plus HX squared so we get 3 h x squared x yep confusing myself now and then we have plus three h squared x and then lastly plus h cubed and again all over h and equals uh cool so again we can cancel that KX cubed that k x cubed we're going to have 3 k h x squared plus 3 k h squared X Plus k h cubed all over H and once again we're going to cancel out H from everything and we're left with finally as H tends to zero three k x squared plus 3 k h plus k h squared and once again if I take the limit as this tends to zero these last two terms will go to zero because we're timesing by a smaller and smaller number so all we will have when we have F Prime of X is simply 3 k x squared as we've done before so this video is a lot shorter um a few reasons for that is that this is a fairly short topic but it's long enough that I don't really want to do the other differentiation with it so when people look up the lesson on YouTube whatever they will be able to find differentiation from first principles and then also plain old differentiation because the differentiation questions can get a lot more tricky so you have like six markers and seven markers again this that's as differentiation A2 differentiation goes even further so that's where we'll leave it for now uh and I hope to see you in the next one