Chain, Product and Quotient Rule: Exercise 3

00:00 Have a look at our next question. We have 2 brackets, y equals to 3x squared plus x minus 1 multiplied by 10x minus 2. Now, here you have 2 options. You can either expand out because it's easier to expand. By expanding I mean you can just times it through. Now usually that's more of a problem when you have some sort of power here. So maybe you have a number and here or there. And that's when we avoid expanding. But even in this case, you can use the product rule and you'll see that it's equally easy. Let's just get rid of that one. It really is up to you. You are at this point where you're learning so many different methods that you have a choice of what to use, but if I just split it into 2 functions. And I say this is function 1, this is function 2 simply because I don't feel like expanding it, we can still apply the product rule to this. So, a quick recap of what the product rule is, dy by dx where the product rule is uv dashed plus vu dashed or the other way around. So, if you've learned as vdudx plus udvdx it's exactly the same thing where the product rule because you have a plus in the middle it doesn't matter what order you write them in. Let's split this so we've got u equals to 3x squared plus x minus 1 and v equals to 10x minus 2 u dashed or du by dx is 6x plus 1. Remember any x just goes to 1 or the number next to it and the constants disappear and then v dash or dv by dx is just 10. Let's put it into the formula so you're literally just cross multiplying but you're kind of going across to multiply. So, we can say that dy by dx now is, if I do vdudx so we have 10x minus 2 multiplied by 6x plus 1. You have a plus in the middle and then it's 10 multiplied by 3x squared plus x minus 1 and it's not dividing with anything because it's not the quotient rule. It's fairly factorized at this point. So we can see if things cancel out by just timesing it out. If you multiply this two, you get 60x squared plus 10x minus 12x minus 2 for the first set. And then here if we times it to 2, we get 30x squared plus 10x minus 10.

02:42 And it's got a few x squared terms that we can combine and a few x terms that we can combine. So I suppose it's a good decision that we expand it out. So, if you bring your x squared terms together firstly, you got 60x squared plus 30x squared.

02:58 10x minus 12x gives you minus 2x and then you also have this plus 10x and then we have minus 2 minus 10. 90x squared plus 8x minus 12. And that's our gradient.

03:14 So, in some sense it's better that you expand it because you get a much simplified answer and there are other instances where it's better that you just leave it factorized but it's just a matter of you either forward thinking or just trying it out on the side and checking whether it's worth expanding out or just leaving it as it is.