So there we have it, the three rules, and examples and their proofs.
So hopefully we're fully convinced and ready to try some of these questions
but before we do this let's just look at a recap and go over all the rules before.
Remember the chain rule, this is the rule that you use when you have function inside of a function.
The first thing that you need to do is spot that function,
so recognize that this is a function within a function and then you apply the chain rule.
We discussed the full definition of the chain rule but also looked at the faster way of doing it
which is the way that we have given here which says that you can differentiate the outside function
as a whole first and then multiply it with the differential of the inside function.
So if you're dealing with the function that looks like f of g of x, where g of x is the inside function,
you differentiate f as a whole first and then you differentiate g which is the inside function by itself.
We then looked at the product rule. Now this is when you have two functions that are multiplying with each other
so once again you need to observe and look at your question.
You need to look at what kind of function it is, whether it's a function inside of have a function
or whether it's two functions multiplying or whether it's both.
You recognize that and then you can differentiate it using the product rule.
So the product rule states that you leave one of the functions as it is,
multiply it with the differential of another, plus leave the second function as it is
and differentiate it with the derivative or the differential of the first.
So you could read that as vdudx plus udvdx where you done them separately,
and then you bring them together into this rule.
The last type of function we looked at is when you have two functions dividing each other.
So you have a function at the top and a function at the bottom.
In order to do that you can use the quotient rule which is a little bit different to the product rule
because it takes into account the function at the bottom,
so you always call the top function u and the bottom function v.
It's really important that you stick to this because if you mix it up this quotient rule
wouldn't be valid for any other combination. So to use the quotient rule you leave v as it is
which is the function at the bottom multiply it with the differential of u
and then subtract the u as it is, multiply it with the differential of v and divide it by v squared.
Let's look at some quick examples now before I let you, have you go at the exercise lecture.
So we're going to try and use all of these three rules,
the faster more efficient way of using them and see if we can find the gradients of the following.