here the point of this is to do two things number one give you another motivation for white chambers awesome and also remind you of what this whole exercise this whole activity is about so just put your pens down and watch it will be more valuable to watch person take that sound later you'll have time so what motivated chain rule in the first place okay or what motivated our us looking at and thinking we need a better tool okay and the reason why is when we had a look at this we thought this this is just ridiculous i mean we could extend it i suppose if you had a lot of time and then you could just go term by term and you could differentiate but we thought just why waste time surely there's got to be a better way okay so that's what we saw here when you look at something like this this is going the other way this is not like arbitrarily more complicated this is actually a very very simple graph but you encounter more or less the exact same problem right which is that well you know it's not quite impractical to expand it's impossible to expand you just can't do anything to it it's as simple as it gets okay but you can see despite being such a different kind of problem chain rule is still exactly what we need to use here right how should we rewrite this the square root of 4 minus x in order to make it clear what we need to do very good so what you've got here is this power okay and once you've done this this is a beautiful thing about how like when i say powerful chain rule is powerful because it just takes such a variety of different things and just make mint speed of them okay i'm gonna do what i said before i'm gonna say well this is why i'm not gonna do my substitution i'm just gonna do the inside and then i'm going to do the outside okay inside then outside so let's have a go at this when i do the inside the derivative of the inside is 4 minus x sorry the inside is 4 minus x so the derivative of the inside is just minus one but four contributes nothing just look at this coefficient minus one there go okay now then i now think i have the outside to deal with that's something to the power of a half we just use our power rule right we bring the coefficient sorry the power out the front to be the coefficient and we reduce the power by one so the power here is a half right so the half comes out the front okay and then what gets gets left in here the power reduces by one it's the same thing we've been doing with all our powers so i've done the differentiation now i just want to tidy it up a little bit being that it was not given it to me in index form i don't really want to give it back in index form if it's at all possible so what am i going to do with this thing this really here with the negative it means go down the bottom of a fraction right you change the sign you cross the line okay so that's going to come with the 2. i've got a minus 1 over that 2 is down there with the square root for minus x okay now let's think about this thing right you could just stop and you can say well i'm done i'm finished i i've got the derivative happy time okay let's just go another step further because the whole point of like me saying we're not just monthly applying rules what is the derivative what does the derivative tell us what's the point of the derivative it's the gradient function right now you look at this thing what does this look like okay what does this look like um you know it's going to be half of a sideways parabola okay but not only that it's not the square root of x it's not even the square root of x minus 4 it's 4 minus x so what's happened to that what effect does that have it's gone that way right it's flipped horizontally because the minus sign is applied to the x which is a horizontal thing the minus sign of course is over here or here it'll be on the y so it would be flipping this way right okay so therefore what does this look like this is the shape okay intercept here four intercept here two okay that's all you need to do it's finished okay what does this thing tell us what does this thing tell us it's beautiful look this square root on the bottom okay square roots are always positive by definition okay that's positive that's positive that's negative so without even graphing it without any getting close to graphene you know the entire derivative is negative right of course it's negative look it's decreasing right not only that where is the derivative defined what is the derivative defined it's defined for x is less than not inclusive because it's on the denominator right less than not inclusive of four correct x is less than four okay what's this thing look like unlike this which has no no denominator okay this guy has an asymptote right there okay can you see why just have an asymptote