00:01
Our next example is now asking us to differentiate 1 over sin x.
00:05
Now think if you can remember what 1 over sin x was called.
00:08
It's cosec of x. But we like to deal with things written in terms of sin, cos, and tan,
so rather than overcomplicating it, let's just start differentiating this.
00:18
So I have y equals to 1 over sin x. If you identified or recognized that this is the same as cosec,
that's brilliant, you're obviously learning your identities.
00:28
But in this case, we can see that we have a number at the top and a number at the bottom.
00:32
You can take you entire sin x up and then just treat it like the chain rule.
00:38
So by that I mean you could do this, sin x to the minus 1
and then it's very similar to the previous example, so you could use it as the chain rule.
00:46
But the idea is to practice the quotient rule so I won't do that.
00:49
We'll just rewrite this as the quotient of u and v,
so I'll say the top value is u and the bottom value is v.
00:57
The quotient rule, dy/dx is vdu/dx minus udv/dx over v squared.
01:06
So we separate it out and then we put it into the formula. So, u equals to 1, v equals to sin of x,
u dashed is zero and v dashed is cos of x. Go straight into the formula,
we're using vdu/dx minus that so dy/dx is just going to be sin of x multiplied by zero
minus 1 times cos of x and then you divide it by the squared
which is just sin x squared or sin squared x. Sin multiplied by zero just gives you zero,
that just goes away, leaving you with minus cos of x over sin x squared. This is done.
01:54
There are lots of other things you could do with this,
so you could write this as minus cos x and then your sin x you could write us,
let's just do it like this, 1 over sin x all squared for now.
02:08
You can see that there are too lots of sin x, so you can write this as minus cos x
over sin x times sin x or sin squared x. You can take 1 bit of cos and sin
and if you remember from the previous lecture, we said that tan x is sin over cos,
so we also said that cos x over sin x is just 1 over tan. So, it's just inverted.
02:39
So, here you can see that we have cos over sin which is just 1 over tan.
02:44
So, we can rewrite this as minus 1 over tan x and then you still have sin x at the bottom.
02:52
We know that 1 over tan x is the same as cot so that's minus cot x
and then we still are timesing it with 1 over sin x and use this knowledge that we did at the start,
sin x is the same as cos x so you can rewrite this as minus cot x cosec of x.
03:16
And what we've just done is found the derivative for cosec x, so if you,
like we said earlier, you could write 1 over sin as cosec of x
and if someone asks you to differentiate that, you'd get this answer similar
to someone asking you to differentiate this because they're both the same thing.
03:35
So, we've worked out that the derivative of cosec of x is minus cot x cosec x.