00:01
Have a look at out first example.
00:03
We’ve been given an equation or a function for drug sensitivity
and we’re calling that R.
00:10
M here is the dosage of the drug in milligrams.
00:13
So we’re looking at a drug sensitivity of a certain drug
and it’s given in terms of this function.
00:19
The question is asking us to find the rate of change of R with respect to M.
00:24
Now, really think about this,
we’ve mentioned this so many times through this course.
00:29
Rate of change with respect to something.
00:32
I’ll take you back to when we looked at the change in Y over change in X.
00:36
How does Y change with respect to X?
And what does that give you?
It gives you the gradient and it gives you dy/dx.
00:44
So with that thinking,
you can conclude that this question is actually asking us
to differentiate R with respect to M.
00:54
Let’s try that out on this function
So given function R of M.
00:59
So that’s just saying that R is a function of M.
01:02
We have 2M square root of 10
plus 0.5M.
01:10
Okay, we’ve discussed that the question is asking us
for the rate of change of R with respect to M.
01:16
So how does a drug sensitivity change with respect to the dosage?
Does it increase?
Does it decrease?
And it’s been given to us in terms of this function.
01:27
Let’s have a look at what we need to do.
01:29
So if it’s asking us for the rate of change of R with respect to M,
we need to do dR/dM.
01:35
So that’s change in R over change in M.
01:39
And that is differentiation.
01:41
So we are differentiating this function here.
01:46
Hopefully, lots and lots of different techniques
are going through your minds right now.
01:50
You’ve got two functions in terms of M.
01:53
You’ve got 2M that’s multiplying with something else.
01:57
You’re starting to think how do you differentiate two functions that multiply together.
02:03
And I hope you’ve concluded that we need to use the product rule.
02:06
Let me remind you of the product rule here on this side.
02:10
dy/dx or a product rule in terms of Y and X
is vdu/dx plus udv/dx.
02:18
So in very simple terms, leave the first function as it is, differentiate the second,
leave the second function as it is, differentiate the first.
02:26
We’ve done this using this formula
and we’ve done this faster when we were doing implicit differentiation.
02:32
To make this a little bit easier, let me just write this as 2M
and rather than the square root, I’m going to write that as a power.
02:39
So I have 10 plus 0.5 M
to the power of a half.
02:44
And here, you can see two clear functions.
02:47
We’ve got this function and we’ve got this function.
02:50
You can call one of them U and one of them V and we can bring them together.
02:54
Or alternatively, we can try and use the faster method
where we leave one term as it is, differentiate the second,
leave the second as it is, differentiate the first.
03:04
So remember that we’re now differentiating with respect to M,
which makes complete sense because this equation is in terms of M.
03:13
I’m going to start with leaving my 2M as it is, so that’s my first term.
03:16
So 2M stays and I now have to differentiate this.
03:21
Look at this closely and decide what kind of function this is firstly
and what method of differentiation you’d need.
03:29
You can see here that you have something to the power of a half
and then you have something inside of that function.
03:35
This is a function of a function
and then you have to think back as to what method you need to use
to differentiate a function of a function.
03:44
We’ll have to use the chain rule method to differentiate the function of a function.
03:49
So the chain rule stated differentiate the outside function first as a whole
and then multiply it with the differential of the inside.
03:58
So let’s try to -
I’ll put brackets around this because this seems a bit more complicated.
04:02
Bring the power down.
04:03
So I’ll bring the half down,
leave the inside function as it is,
and decrease the power by one.
04:12
Don’t forget now to multiply it with the differential of the inside.
04:17
So the inside says that you have 10 plus a half M,
differential of that is just going to be a half or 0.5,
however you prefer to write it.
04:26
So the differential of the inside,
I’m writing as half because I already have a half there,
it’s easier for me to combine it.
04:33
That’s the first half of the product rule then.
04:35
The second half, remember that the first bit I left this term as it is
and I differentiated the second.
04:42
I’m now going to differentiate this term and leave the second term as it is.
04:46
So the differential of 2M with respect to the M is just 2,
and the second term can stay as it is, we don’t have to fiddle around with it.
04:54
That can just stay as it was.
04:57
And so we’ve used the product rule to find dR/dM
or the rate of change of R with respect to M.
05:05
It obviously isn’t very tidy, so we just need to do some algebra to tidy this up.
05:10
I can multiply those two numbers together.
05:12
So half and a half.
05:13
So I’ve got 2M
and then I’ve got a quarter on the inside
and then I have 10 plus 0.5M to the minus half.
05:22
You can take this down if you want
or you can leave it for a minute and then we’ll take it back in the next step.
05:28
Here, you can rewrite this as 2
and you can change this to square root of 10 plus 0.5M.
05:36
These two numbers will cancel,
so the 2 and the 4 cancels to give you 2,
giving you M over 2.
05:42
And then like I said, you can bring this down.
05:45
So because this is to the power of minus half,
you can write it as positive half ,
as 10 plus 0.5M
plus 2 square root of 10, plus 0.5M.
05:57
And what you’ve just found here is dR by dM
or another way of writing this is just R differentiate it with respect to M.
06:07
So if I’m using the similar notation to what they’ve used in the question,
they gave us R as a function of M.
06:13
If I differentiate it, you can write it as R’M or dR by dM
and we’ve differentiated it.
06:19
So really, you found a function which represents the gradient
of how drug sensitivity is changing with respect to dosage.
06:31
If we now look at answering the second part of the question,
so it says find the rate of change of R with respect to M,
which we found here as an expression.
06:40
It then says find R’ of 50.
06:43
So it’s saying when you put 50 mg into this equation,
what is the drug sensitivity just as a number?
So we’ll do that here on the side, it’s just a simple matter of substituting numbers.
06:56
We have R’ of 50.
06:58
And all we’re doing is changing our M, which was here, to 50.
07:03
So we’re using exactly the same equation.
07:05
So we’re using this equation and replacing all our Ms here with 50.
07:10
So I’ll end up with 50 over 2,
root of 10 plus a half times 50.
07:20
And then I have 2
root of 10 plus a half times 50.
07:28
If we continue just simplifying this out,
the 2 and the 50 cancels out to give you 25.
07:33
A half times 50 gives you 25 plus 10, so I have root 35
and then I also have 2 root of 35 here.
07:42
You can take root 35 as a common denominator,
so remember when you multiple root 35 by root 35 in the enumerator,
it's such going to give you 35, so this gives us 25 + 2 x 35 all over root 35,
which when you add give you 95 over root 35.
08:00
As this doesn't simply you can leave this as it is or work it out as a decimal on your calculator?
Therefore the value R dash at 50 is 95 over root 35.
08:13
And then you can do all types of mathematical analysis on those results
and you can try and change the dosage and see how drug sensitivity varies.