A Level Maths Made Easy - Quotient Rule Differentiation

hey guys welcome back to another a Lev Mass made e easy video this time we're looking at the quotient rule for differentiation the quotient rule is used when you have a quotient which is essentially when you have two functions of X dividing each other then the quotient rule is very very very ugly and it's very particular so in the product rule it didn't matter which way round we multiplied the things right we could do V * D udx or we could do U * dvdx add either way around but in this case because we're subtracting it does actually matter which way around you call u and v v has to be the function on the denominator now I'm actually going to show you using the very first question that you don't actually need to use the quotient rule ever and in fact the quotient rule the way you derive it is you simply use the product rule and the chain rule together in order to get this expression here so for the first question I'm actually going to prove it to you I'm going to use the quotient rule of course but I'm also going to just use the product rule because I think again in my opinion instead of trying to memorize this disgusting formula that's very particular because it's V minus U you can't do uus V and it's over V ^2 not over u^2 that it makes a lot more sense to use the product rule and the chain rule because those can be done in any order so if we use the product rule for something like this I'm going to change F ofx to just y because it's a bit easier then I'm going to rewrite this as x * x + 2 theus1 using those laws of indices then if you remember the product rule we're going to have U is equal to x v is = to x + 2-1 we're going to differentiate U we're going to differentiate V and remember this is the chain rule but remember the shortcut I showed you what you're going to do is you're going to bring the power forward so it's negative you're going to differentiate what's on the inside so X+ 2 differentiated is just one so it's minus one so I'm not going to write the one then x + 2 to the minus 2 then all you do is you cross multiply so we have these two times each other so we have x + 2us one - x x + 2us 2 now that would be enough for Dy by DX okay that would be enough but as you're going to you're probably going to be able to tell when we use the quotient rule we're not going to get something that looks like this we're going to get something that looks like this so to prove to you that it is indeed the same what I'm going to do is I'm going to write this in a slightly different way so we have 1/ x + 2us x/ x + 2 2 and then you can combine those fractions by timesing the first one by x + 2 top and bottom so we get x + 2 - x over x + 2 2 and then finally x - x is just 0 so we get 2 x + 2 2 now we can use uh the top of this we're going to use the top of this in fact so we're going to use this first part but this time we're going to use the quotient rule so the quotient rule says V * DX so V is this is the denominator here uh which is just x + 2 in fact Ah that's a really good point all right so we're going to have to kind of re redefine this slightly so U is just equal to x v is just equal to x + 2 then we have DX is 1 DV DX is one so now if we do exactly what it says V * DX that's going to be V * DX just x + 2 then it's minus UV DX so minus U * this and minus X so you're probably seeing that it's the same numerator we got here all over v^ s so that's going to be x + 2^ 2 which again if you put that all together we get the exact same answer using the product rule as using the quotient Rule now you might say well this looks a lot quicker so why would I use it look if you want to use it that's totally fine but the problem that I see with a lot of students is mixing this up it is really really specific with the product R and chain RW I could have multiplied these in any order I could have moved them around and because I'm adding them it doesn't make a difference so you have to keep that in mind whereas this you have to do vdx minus UD vdx you have to and you have to divide it by v^2 again if you think you can remember that I'm just going to remind you that in an exam it's very easy to make mistakes with this so I would recommend just using the product rule and the students that I've taught after a while they just say well I might as well just remember the product rule and the chain rule product rule and chain rule really easy to remember quotient rule isn't now while the quotient rule is quite quick it is a shortcut and it is a shortcut all it is is the product rule and chain rule simplified so I would recommend just using the product rule personally so would recommend just doing these as a bit of practice uh for yourself but I'm just going to jump straight into them again I'm going to use the product rule and the chain rule something that would be quite useful for you if you plan using the quotient rule is try and use the quotient Rule and get the same answer as I do but again remember that I'm writing it in a different way than you might expect now if they do say you have to express it as a fraction you would have to combine the fractions and simplify but remember this is the correct answer as well but if they say express it as a simplified fr fraction you have to do the combining that's only really relevant if you're using the product rule and chain rule if you're using the quotient rule it does it for you so that should hopefully give you enough time to you know pause the video and try for yourself again you can use either method that you so well choose again differentiating is a half xus half and DV DX remember you bring the power forward so that makes it negative differentiate what's on the inside that gives you three the number three and then you times by the same thing but you subtract one from the power and then all you're doing is you're cross multiplying so the first thing I'm going to have is for dydx a half x- half * 3x + 2^ minus one and then it's technically going to be a minus because we have a minus here timesing those two we get 3 x half 3x uh + 2- 2 and again that is the correct answer but if you wanted to simplify you would again you would have to times the first one by top and bottom by 3x + 2 and combine it in that way or again you could just use the quotient Rule and get there as well doing the same thing over here again we have U = 2x + 1 we have V is = x - 3 to the^ of minus a half it's minus a half because eight it's on the bottom of the fraction so it's negative and it's a half because it's square root differentiating both of those gives you two and again bring the power forward so it's - half differentiating that just gives me 1 so it's just- half on the outside x - 3^ -3 / 2 and once again cross multiplying so so yeah I do personally prefer using the product rule in the chain Rule and if a question does ask me hey you need to leave it in terms of a simplified fraction algebraic fractions is something you need to be good at for a Lev maths anyway so it's good practice for that combine them and you do get the exact same answer I know I'm hammering this home quite a bit but it is very important that you don't make these uh kind of simple mistakes because it can cost you right and at the end of the day every single Mark is useful most people are within five marks of getting the next grade so if you think about every single Mark that you lose to silly mistakes that could have been what gets you to the next grade up unless you get an A star of course because there is no high grade so the people that get A's are normally within five marks of getting an a star people that get B's are normally within five marks getting an A and so on so you do you should kind of keep that in mind so again while I'm not opposed to using the quotient Raw it's whenever we do tests in school or kind of with my tuition students sometimes if they use the quotient rule they make a mistake on it and then they just go back to using the product for chain roll because you know you get practice with that anyway right and it's super super easy for you as something to finish off with because again I do I there are loads and loads of questions on this but I don't want you getting too bored I wanted to show you how you can prove this so this was actually technically a show that question and I want you to think about how you can do this because this again is quite useful now you again you could use the quotient rule you could use the product rule you could use whatever you want but with this how would you show this well it's actually not too bad if you think about it the left hand side is actually sinx over cos x which again depending on if we want to use the quotient rule or the product rule in fact maybe I'll use both just to show you how you can do it let's start with the product R and chain rule just because again that's my uh personal fave cos x theus1 then all we're going to do is we're going to say that U is equal to sin x uh V is equal to cos xus one now the reason why I'm putting this in Brackets is do not confuse this for the inverse cos it's not the same thing it's cos x to theus1 it's not cos -1 x right that's why we start using the notation of Arc C and arc sign instead of saying Co Theus one we say AR C because it is very confusing when you're dealing with reciprocals of cosine functions instead of inverse functions so uh differentiating this when you differentiate sign it just becomes it stays the same sign and it becomes cos x and when you differentiate this again remember you bring the power forward so that's a minus one then when you differentiate this you get negative sign so it's swap sign so if it's negative cos it becomes positive sign if it's cos becomes negative s so that's sin x so those two are going to cancel out then it's cos x to Theus 2 so if I simplify this we just get sin x cos x to the -2 and then again we're going to multiply and you know cross multiply and all that stuff so the first thing DX is cos x time that well you can probably see that that's going to cancel out but you know I'm not going to not going to make too crazy then it's going to be add we get sin x time this so we get sin 2 x time cos x- 2 that looks fine so that becomes one plus let's call that sin^2 X over cos 2 x just to make it a bit easier which is 1 + 10^ 2 x now again that isn't the answer that they've got here but if you remember when you have sin 2 + cos s = 1 you can divide through by either s or cos to get certain different values right so if I since I want T squ I'm going to divide through by cos squ by the way I should have an X after this I'm just being completely honest with you so that makes it 10^ 2 x + 1 = 1 / cos 2 is SEC s x so this thing here is equal to SEC SAR X which is what we were supposed to show in the first place now when we do it using the quotient rule you're going to notice something a bit interesting which is we don't actually need to use this last bit which is very fun so if I go back up to the top just so you guys can follow along I'm going to copy the quotient rule down uh let's leave it over yeah let's leave it on top that makes sense so in this case we have we're still going to write tan as Sin / cos so we get U is equal to sin x v is equal to cos x remember we don't need the minus one because this is defined as V being on the bottom on the denominator because it's defined as that we don't need to make it cos to Theus one then we get the same type of thing D DX isal to cos x DV DX is equal to sinx and and we're just going to sub it into this formula now so we have V DX so cos x * cos x that's cos 2 x minus U sinx * okay never mind that's actually plus now actually no let's leave it as this minus - sin 2 X over v^ 2 which is cos 2 x so we get that becomes a giant plus right sin + cos is 1/ cos x which is equal to s x so again it's just another way to get to the same kind of result um which is interesting right like it's it's interesting that there's two different ways we did use the sin square plus cos squ equal 1 so maybe I should actually write that so people don't get confused whereas in the other answer we used the tan^ 2 x + 1 = SEC s which again it's different right it's nice it's interesting um maybe I'm the only one that actually thinks it's interesting but that's totally fine but again we've basically shown the result that the derivative of tan X is equal to se s x and this just so you're aware back in the olden days you know back when I did the a levels in the 1800s when we differentiated this we actually got the result that the differential of 10x is se s x and some people were some s were rightfully complaining about year on year they give fewer and fewer different differential results right so they give few and few of these and the exam Bo's reasoning is that technically speaking you should be able to get from anything you differentiate to the actual answer this actually includes sinx and cosx so in year 13 you expected to be able to differentiate sinx from first principles using the small angle approximation same with cos x and then using that you can differentiate tan using the quotient rule or the product and chain rule so technically speaking the reason why they don't give you all of these results is because you can derive them yourself I don't like that personally because I don't think because we're doing maths we're not doing further maths we don't need to derive everything and not just that the exam is quite short in terms of time so I don't know if it's particularly Fair because they should give you more time in the exam if they expect you to derive everything yourself but that's neither here nor there uh here's how you do it and again that's basically it for the quotient rule you don't really need to use it you can use the product rule in chain rule as I've said about 30,000 times in this video but next we're moving on to integration which is where things get a little bit harder um well I mean integration is coming on a bit later I think next is actually differential equations which again it's a lot harder it's a lot more fun it's also a lot more applicable to real life and to a lot of what your careers are going to be if you're going to be an engineer physicist um going into medicine anything like that this is going to be a bread and butter economics as well