we're going to talk about primitive functions now underneath here what i want you to do is um leave a space underneath here maybe a couple of lines and then after a couple of lines i'd like you to write f of x okay in calculus we've been talking about functions that's kind of our main object and we apply the process of differentiation to a function when you differentiate when you differentiate the object that you get out of that goes by a couple of different names at least what kinds of names would you call it you differentiate and the result is the we could call the gradient function the gradient function that's why this is useful i'm going to write that down in a second um we would also call it like one word the starts with the d the derivative right i'm going to write that down so we call this the derivative and in this scheme we would notate it as f dash we call the derivative because it comes from the function that you're interested in right so derivatives or of course the gradient function they tell us something about the original function and how it behaves or how it looks like okay now it's therefore not that much a stretch of the imagination to imagine well what if this function itself was a derivative you know maybe this is like a first derivative and this is a second dude that means it also came from something before it does that make sense there's like an extra ladder that's why i asked you to put a couple of um empty things now if you put the arrow up right from f of x you can get f dash right but where did f x come from there's a few different ways to notate this but the most common way in this scheme is with a capital f okay you can read this in a couple of different ways like this is f dash this is sometimes called f prime but the most common name is that it's called the primitive function prime primitive you get the idea you can see derivative means okay what do you get out of this primitive means where did you come from like primitive human species and that kind of thing okay now in here what is the process by which we get up to this okay now please note some of you have learnt ahead and you might think you know what belongs here what word belongs here i want to be very careful with this and very deliberate in this scheme in this context with the knowledge that we have now because what we're doing is the opposite of differentiation the opposite of differentiation this process is properly named anti-differentiation as it were you're undoing the process of differentiation right if i were to differentiate f prime if i were to differentiate the primitive you would land on this function the one that we started with okay anti-differentiation is what it's called we'll come to the other names that that process might get to when it's the right time okay now here's the weird thing this is all abstract okay we want some actual examples of what's going on so in parallel to this let's just come up with an example a simple one like say x squared okay now we know what happens when we differentiate if we differentiate this then of course the derivative of x squared is 2x all good okay now think about this when you differentiate what you're doing is you bring the power down and you reduce the power by one so being that we're trying to undo this we're trying to anti-differentiate if i think about all right where did this come from what was it that got differentiated to become x squared i have to do the reverse of that now please note let's actually write this down okay the the rule that makes this work in our head is bring power down make it the coefficient right and then reduce the power by one okay so when we do this anti-differentiation process we have to do this exact thing but everything is backwards everything is backwards so there's a step one step two right do you notice when we differentiate you have to do it in this order you can't reduce the power first and then bring it down because that's not the derivative okay so i have to do this order differently so i'm going to do this step first but i'm also going to do it in reverse so instead of reducing the power i'm going to increase the power all right so that's the first thing i do i'm going to increase power by one okay and then now i'm coming to this so instead of bringing the power down right which is in a sense multiplying by that power what am i going to do here what's the opposite of multiplying dividing right in fact a better way it's just so colloquial that we're very comfortable with it a better way to say bring power down is multiply by the power here we're dividing by the power does that make sense okay let's do it the power is currently two so when i increase it you're going to get three that's the first step now i'm going to divide by this new power which gives me a third now one of my favorite things about this part of mathematics is that it's so easy to confirm just like solving an equation did i get it right because of course if i anti-differentiated correctly then differentiation should me bring me back to here let's just quickly see if it works multiply by the power which gives you 3x cubed on 3 so the 3s will cancel that's good and then reduce the power by 1 which comes to 2. thumbs up it worked okay ah but here's the weird part and here is why my title and your title has a plural on it okay functions have derivatives one function has one derivative and only one like this has to be the answer right there's no other derivatives that possibly be but when it comes to primitives there are actually many other things that we could have started from that when you differentiate will still land you on x squared think about this right if i had something up here like say x cubed on 3 that stuff is all still the same and if i had something over here that when i differentiated it didn't add anything or subtract anything didn't change anything you'd still end up with x squared wouldn't you well what might be something else we could have there plus five why not you differentiate that add zero no big deal or in the same way i could say x cubed on three i could subtract something right subtract anything you like something weird right it's still going to differentiate and come out in the wash you still will end up with x squared okay so what you're creating up here maybe you want to put this in another color or something like that what i've got up here and it's this there's many more right what i have up here is what we call the family of primitive functions it's a family they all share the same dna but they can be different okay now because i've just got examples here right i'd much prefer to actually generalize this a little bit i can add or subtract any number i like and it will still be in the family okay so therefore what we say is well if you're adding or subtracting any constant you like let's just give the constant a name constant constant what would be a good letter to denote a constant well it's interesting i mean we choose sometimes completely nonsensical letters but whatever uh thankfully here it makes sense we use the letter c for the constant so we would say that everything in this family of proven functions is of the form wrong color is of the form x cubed on three plus c okay so that constant can be whatever anything you want let's label that accordingly okay now underneath here uh and with this we'll use this example let's just quickly draw what this might look like okay x cubed on three if it's just x cubed because on 3 is just a change of scale right what does x cube look like how would you describe that what kind of features does it have that you know about x cubed has a point of inflection and it's not just any point inflection it's a horizontal point of inflection so let's go ahead let's draw it if i put some coordinates for scale then that would make it x cubed on three now what would these guys look like in terms of how they look different to this well the x cubed on three plus five where's that in relation to this it's just been shifted vertically so you can translate it okay so i can draw the same thing just higher like so and that would just be five there what about the other one x cubed on three minus pi it's the same thing but lower okay so i'm just going to draw the same theme under here and i guess that would make that negative pi because that's weird all right so why is this whole family going to end up at the same thing well if i were to take any spot let's see how well i've drawn this i don't know maybe quite badly if i were to take any spot on this curve pick an x value any x value x equals 1 for instance if you have a look at all those spots for x equals 1 here here and here at each of those spots if my drawing is reasonable the tangents will all be parallel do you agree if the tangents are all parallel that means the gradients are the same because they all have the same gradient function does this make sense it doesn't matter how far up or down you are you're going to get the same gradient all of these things belong in the same family does that make sense so for now this is the end of this topic that we know how to differentiate but we also know how to come to come back to where did you where did this come from right what was the primitive that preceded this okay if you are ever asked for the primitive function uh for this you would say x cubed on three plus c how about just to make sure you don't get away with well sticky how about something like this there we go there we go just for the sake of what's this thing going to be what function did this come from well i know how to differentiate it to anti-differentiate and do everything in reverse let's go one at a time tell me what to write three x to the power of you've increased the power by one and then what do you do with that new power you've got to divide by that six so it's going to end up on the denominator yes so far so good all right this guy think because this is an x to the zero term isn't it so you're going to increase that power by 1 which gives you x to the power of 1 and then you're going to divide by that new power right of course i can tidy this up a little bit this of course will be well let's see that's going to give you 2 so x to the 6 on 4 plus 31x that's only one of the primitives right that's one possible primitive so how do i turn this into the primitive function this constant right aha now i've covered myself this could be anything does that make sense okay