We've done a lot of calculus already, we've looked at differentiation from first principles,
we've looked at the easier way of differentiating,
we've then moved to lots of different types of functions.
So we looked at applying the chain rule when you have a function within a function,
we looked at the product rule when you have functions multiplying,
we looked the functions dividing and how to use the quotient rule.
We've done lots of proofs so easy and difficult proofs
and we're moving deeper and deeper in the world of calculus and by the end of this course,
you will all be expert of calculus. So the next thing for us to go over is a trig function,
or trigonometric functions and we're going to learn not just the identities
but also how to differentiate them. So we're going to start to put them into our equations
and start to differentiate them. Now this is fairly useful for any field really
but particularly for medicine where you are measuring graphs
that perhaps follow the sin, cos or tan pattern.
It really as quite important to understand how to find the gradients
which then allows us to predict further patterns.
So let's go straight into it and start to look at differentials of set functions.
We'll start by looking at this table, I'll give you some of the differentials of standard functions
and then we'll look at deriving some of them just to build up on our algebraic skills,
on our analysis skills and then we'll go straight into doing some questions.
So these are some things that we just have to learn but I will show you the proof just to convince you.
If you have a function sin of x, the gradient or the differential of sin of x is simply cos of x.
If you're differentiating cos of x, the gradient or the differential of cos of x
is given by minus sin of x, this can get a little bit confusing but it's just a matter of practicing.
Tan of x differentiates to sec squared of x and we will define what sec is shortly.
We have some further functions which we can also put into this table
and you can see as we do questions, how to apply them within your differentials,
so you have cot of x which differentiates to minus cosec squared of x.
We have sec of x that differentiates the sec x, tan x and then you have cosec of x
which is minus cosec x, cot x. Now the last three functions here,
you can always derive yourself. So once you know the basic skills
of how to differentiate and how to differentiate sin, cos and tan.
The rest of them you can actually just do from scratch.
So you don't have to learn them, we just need to learn the skills
that helps us differentiate any type of function.