So, let's actually start doing some numerical examples using sin, cos, and tan.
Let's look at our first example here which asks you a nice straightforward question
and it's asking us to differentiate y equals to 3 tan x.
All we need to remember is what does tan differentiate to.
So you can look it up in your notes to start with but eventually,
we should be able to know these by heart so we can at least differentiate sin, cos, and tan,
without having to look them up or derive them.
So we have our function y equals to 3 tan of x. It's no different from general differentiation.
You just have to differentiate and you have to differentiate this
but this time we use the rules, so we know what tan differentiates to.
So when we differentiate this, dy/dx, the 3 is just a constant so that can stay there,
and then you have tan x which we want to differentiate
and remember that tan differentiates to sec squared of x as we've just arrived
or we've seen in the rules at the start, and that's simply the answer.
So there's nothing more to it, all we're doing is changing these trig functions
to their standard differentials, if you'll know them, it just takes less than 5 seconds
to differentiate something as simple as 3 tan x to 3 sec squared x.
So let's build this up now. We now look at our second example
where we're now dealing with a cos function but there's something a little bit more
interesting about this so let's just write this out. We have y equals to 5 cos of 3x,
notice now that this is a cos function, but it's not just a cos function,
it's cos of a different function. Now, do you remember when we looked at functions
such as x plus 3 to the power of 5, and we said that that is a function of a function.
So you have a big function and you have a little function. Apply that thought to this.
We have a big function cos and then inside of cos, we have another little function,
it's not just cos x because you know that cos x just differentiates to
when dy/dx goes to minus sin x. However, this is cos of 3x.
So we're going to have to apply the chain rule now but this time to trigonometry.
So if we are differentiating this, we have dy/dx, the 5 is just a constant that doesn't bother us
so let's just leave it there. Now, let's try and differentiate cos of 3x.
Remember, chain rule, big function first, leaving everything on the inside
and then you multiply with the differential of the inside function. So, let's attempt that.
Cos differentiates to minus sin. So we just write that as minus sin of 3x.
Remember, we're not changing the 3x and then we multiply it with the differential of the inside.
So the differential of 3x is just 3. So now it's just a matter of tidying this up.
We can multiply the 5 and the 3 together and the minus, so that gives me minus 15 sin 3x,
and it's really that simple. The idea is that when you look at these questions
before you start to worry about what is actually happening,
you just decide what kind of function it is and then you do it step by step.
So if you spot a function of a function be it with something as simple as brackets
or with sin, cos, or tan, you just have to use the chain rule,
look at the whole function as one and no matter how complicated the inside is,
you then leave it as the first step and then just deal with it
after you've differentiated the outer function. So, outer function first
and then you take the inside function and then times it with the differential of that.
Next example, we're now looking at y equals to 15 sin of 2x squared plus 1.
So let's talk about what kind of function this is firstly. I have y equals to 15 sin of 2x squared plus 1.
Okay, so it's fairly similar to what we just did.
We have a constant on the outside which doesn't bother us. We have sin of something.
So we have the sin function and then inside, we have another function.
So we have a function of a function and we will need to use the chain rule
along with knowing what sin differentiates to. So let's start differentiating this.
We have dy/dx, like we said 15 can stay, doesn't bother us.
Sin of anything will go to cos of anything so we leave that as it is.
So sin of 2x squared plus 1 goes to cos of 2x squared plus 1 but it's not over yet
because we also need to multiply it with the differential of the inside.
So the differential of the inside of this function is 4x. So remember bring the power down,
decrease the power by 1, and the plus 1 just disappears.
So we can multiply this together. We can do 15 times 4
so that gives you 60x multiplied by cos of 2x squared plus 1.