In this video, we're going to be looking at how to differentiate
very simple functions from first principles. So to begin with,
what we want to have a look at is a straight line.
A straight line has a constant gradient, or if we prefer its
rate of change of y with respect to x is a constant.
If we take the straight line y equals 3x + 2,
we can look at its gradient.
We take two points and look at the change in y divided by the
change in x.
When x changes from -1 to 0, y changes from -1 to 2,
and so the gradient is 3.
No matter which two points on the line that we use, the value
of the gradient is always the
same. It is always 3.
Now we can see that in another way, by looking at a table of values.
So let's take x ranging in values from -3 up to 3.
And let's have a look at 3x.
Which is simply the values of x we've got multiplied by 3.
And then add 2 to those values to give us the function
y equals 3x + 2.
Now as we look at this table of values, what we see is that
for every unit increase in x we always get 3 units increase in y.
So as we go from -3 to -2, y goes from -7 to -4.
As we go from 0 to 1, y goes from 2 to 5.
For every unit increase in x we get the same increase in y of 3 units.
And so the gradient of this straight line is 3 or
the rate of change of y with respect to x is also 3.
Now, nice as it is that a straight line has a constant gradient
and that y is changing at a constant rate with respect to x,
this is not always the case. So for instance when
you are in a car and you press the accelerator, you watch the
needle on the speedometer. The needle doesn't go up gradually,
it doesn't go up constantly. The rate of increase of the speed varies.
Similarly the acceleration due to gravity, now near the surface of the Earth,
we assume that it's constant. But Newton actually showed that
it was proportional to 1 over r squared, where r was the
distance that we were away from the surface of the earth. So we
can see that as we move further away from the surface of the
Earth, the acceleration due to gravity gets less, but it
doesn't change constantly. It doesn't change uniformly.
So to see what's happening we need to look at some very simple curves.
For a curve, we can see that if we join any pair of points, we
get a straight line.
But when we join different pairs of points, we tend to get very
different straight lines with very different gradients.
We can see this again by looking at very simple curve y equals x squared
and looking at a table of values for it from -3 again
up to +3. So now let's calculate the value of
the function y equals x squared at each of these values.
Now let's have a look how the value of the function is
changing as the value of x changes. So if we have a
unit increase in x from -3 to -2, we find that this time
the value of y is gone down from 9 to 4, so it's gone down
by 5 units. Not a problem if that decrease is going to be
sustained. But when we look at the next unit.
From -2 to -1, we find that it's gone down from 4 to 1,
y has decreased by 3.
And in the next unit it is decreased by 1, but then it
starts to increase again so we can see that for a very, very
simple function like y equals X squared, y is not changing
constantly with respect to x. Now this is not a problem so
long as we can find a way of
calculating it. To start that process, let's just have a look
at a sketch of y equals x squared.
And let me pick this point on the curve.
Now as we look at that look at this very tiny little bit,
that's almost straight, and if I was to home in and magnify, it
would be almost more straight and it would appear to be
straighter and straighter so as we homed in closer and closer to
the point what we would see would be much more like a
straight line. What straight line might we be able to take
to represent that sort of local straightness? And the
answer would have to be a
tangent. Through the point.
And that gives us what we need, because this tangent is a
straight line and we know that a straight line has got a constant
gradient, and so this is the definition that we make.
That the gradient
of a curve
y equals a function of x
at a given point
is equal
to the gradient
of the tangent
to the curve
at that point.
So notice we haven't defined the
gradient of the curve globally, so to speak,
we've said that at any particular
point, the gradient of that curve is going to be the same
as the gradient of the tangent. So now what we want
to see is how can we use this definition to help us actually
calculate that gradient at any particular point?
Let's take a fixed point P on our curve and a sequence of
Points Q1, Q2 and so on. Getting closer and closer to P.
We see that the lines from P to each of the Qs gets
nearer and nearer to being a tangent as the Qs come
nearer and nearer to P.
So if we calculate the gradient of one of these lines and let
the point Q Approach P along the curve than the gradient of the
line should approach the gradient of the Tangent.
And hence the gradient of the curve.
So let's now try and set up a calculation for the curve
y equals x squared at the point x equals 3, so that this
calculation represents what we've just seen.
So first of all, I'll sketch the curve.
So there is y equals x squared.
Here will take our point.
Where x is equal to three. I'm going to take a point Q,
which is near to P on the curve and I'm going to join up
those two points with a straight line.
So there's my point, P as my point Q and I'm going to take
a triangle formed by dropping a perpendicular down to reach the
horizontal through P. So that will give me a little right angle
triangle with the right angle at the point are.
Now I'm going to take small values of PR.
So that I get the coordinate
of Q. So let's line that up. PR will be first of all 0.1,
then 0.01, 0.001
and finally 0.0001.
So what will be the x coordinate of Q, x coordinate of Q?
Well, it will just be 3 + whatever PR is
so that will be 3.1, 3.01, 3.001 and 3.0001.
Now I've got the x coordinate of Q, let's calculate the y coordinate of Q.
So that means we've got to square each of these,
so our first result will be 9.61 when we square that.
Next is 9.0601.
Next 9.006001
and the next 9.00060001.
So what's the change in y for each of these?
In other words, what's QR?
We know that the base for where P is is if x is equal 3 at P,
y must be equal to 9. So QR, the increase in y
we can get by taking 9 away from each of these.
That's 0.61, 0.601, 0.006001, 0.00060001.
So now we need to calculate the gradient of
each of these little segments PQ. So what is that gradient?
will it be the increase in y, which is QR, over PR, which is the increase in x.
So we've got 0.61 divided by 0.1. So to do the first one
I'll write it down 0.61 divided by 0.1, that is 6.1.
If we do the same calculation here, the answer is 6.01.
Do the same calculation here and it's 6.001.
And do the same calculation here and it's 6.0001.
So let's just have a look what's happening here. This gradient
seems to be getting nearer and nearer to 6. And 6, perhaps by coincidence
is 2 times by 3, where 3 is the x value at this point, P.
So we need to do this calculation again, but we need
to do it in general.
So let's set that situation.
We take our axes.
And we plot our curve y equals x squared.
Take my point P on the curve and P is (x,y).
And I take a point Q a little further along the curve.
How far along the curve is it?
Well, I'm going to increase x by a very small amount.
So we got the same triangle.
As we had in the calculation.
So there's R.
And I am increasing x, to get to R, by a very small amount and
that small amount is going to be written as delta x.
Now I cannot emphasize enough that delta x is not delta times by x.
It is a single symbol on its own, and it represents a small change in x.
What's that resulted in? Its pushed the point further along the curve to Q,
and so we've got a change in y of delta y
So that the coordinates of Q are now x + delta x and y + delta y.
Now. We've got the diagram labeled up.
And so what we need to be able to do is do some calculation.
What do we know? We know that this is the graph of the curve
y equals x squared. So here at the point Q we know that
y + delta y is equal to x + delta x all squared.
This we can multiply out. It will give us x squared plus 2x delta x,
and I'm going to put that in a bracket to emphasize the
fact that delta x is a single symbol,
plus delta x all squared.
But what else do we know? Will we know at this point P that y is
equal to x squared.
So y + delta y is equal to this but the y part is equal to x squared.
So that tells us therefore that the change in y, delta y, is this bit here:
2x delta x plus delta x all squared.
So let's now calculate the gradient of this line PQ.
The gradient of PQ,
well, it's QR over PR
or the change in y,
delta y, over the change in x.
And we've got expressions for both of these:
delta x is delta x and delta y is this.
So we've got delta y over delta x is equal to...
Let's remember that this was 2x delta x plus delta x all squared
and that was all over delta x.
Now, delta x is a common factor here.
Let's take it out as a common factor: 2x + delta x
times by delta x; all over delta x. Now delta x is a small positive
amount of x, so we can cancel it out there and there.
So we have delta y over delta x is equal to 2x + delta x.
Now remember what we're doing.
We've calculated the gradient of this line PQ and we're going to let
Q come nearer and nearer to P along the curve.
So we're letting delta x tend to 0,
until this secant or cord, PQ, becomes the tangent.
And so we can say that the gradient
of
the tangent
at P is equal to...
Now we have to use some mathematical language.
What happens to this expression as delta x approaches 0?
Well, it would seem that if delta x got smaller and smaller
and smaller and smaller and smaller, it got to zero.
This would become 2x. We are approaching a limit of 2x and so
we use some mathematical language to help us say that. So
we say that this is the limit,
as delta x tends to 0,
of delta y over delta x
is equal to
the limit as delta x tends to 0
of 2x + delta x
And if delta x is going off to 0, this is just 2x and notice that
agrees with what we had before. When we said that the gradient
was 2 times by 3 equal 6 at the point where x is equal to 3.
Now we could do this calculation in just the same
way for all kinds of different curves, but it
would become a bit of a bind if we had to keep writing
this down all the time. So we have a special symbol for
this phrase, the limit.
as delta x tends to 0 of delta y over delta x.
And we write that as dy by dx and we say d y by d x
and that's a symbol all on its own as a unit which stands for
the limit as delta x tends to 0 of delta y over delta x,
and in the case we've just
looked at, y equals X squared, dy by dx we've just seen is equal to 2x.
Now. Another video in the sequence will show how this is
done for y equals sine x and the processes that we go through.
We often use function notation y equals f of x.
So in function notation,
our point P, which was the point x, y,
becomes the point x, f of x.
Our point Q, which remember was x + delta x, y + delta y
Well, this remains as x + delta x for the x coordinate,
but the y + delta y. Well that's the function of x + delta x.
And so if we want to have a look at what we've done before, if
you remember, then we take
the change in y, that's delta y. So the change in y is:
what we had here, at Q, minus, what we had at P.
So that's f of x + delta x minus f of x, and so our gradient delta y over delta x
is f of x + delta x minus f of x all over delta x.
So dy by dx
is the limit as delta x tends to 0
of the change in y divided by the change in x that gave rise to it.
Which is the limit
as delta x tends to 0
of f of x + delta x minus f of x all over delta x
So, if we are using the language of functions. That's the
language that we use and sometimes for functions when
we're differentiating them instead of using the symbol
dy by dx, sometimes use the symbol a dash to indicate
that we've differentiated with respect to x.
Let's just take one example using this
notation, f of x is equal to the function 1 over x.
Have a sketch of this curve for positive values of x.
There it is coming down like that. So if I take
a point P and I increase the value of x by a small
amount to give me a second point Q on the curve. Then
I can see that in fact,
there's my delta x.
And there's delta y, so for all delta x still means a small
positive increasing x, delta y just means the change in y and
we can see that this is actually a decrease. Let me complete.
This triangle by drawing in that secant, PQ and let's
have these coordinates, this is x, y.
And this is. x + delta x
and then f of x + delta x.
So let's have a look what we've got: delta y is the change in y
and so it's f of x + delta x minus f of x.
And so delta y over delta x is
f of x + delta x minus f of x, all divided by delta x.
And this is the gradient of this line PQ.
OK, what is f of x + delta x well, f of x is 1 over x so
this is 1 over x + delta x.
minus f of x, which is just 1 over x,
all divided by
delta x.
Here we have two fractions and we are subtracting them, so we need a
common denominator and that common denominator will be
x times, x + delta x.
So I've multiplied this by, this x + delta x, I have
multiplied it by x, so I must multiply the 1 by x.
Minus, and I've multiplied the x by x + delta x, so I must
multiply the 1 by x + delta x and then this is all
divided by delta x.
So let's look at the top, x - delta x, minus delta x.
So I've got x - x that goes out.
And I got on that line there - delta x.
All over, well, I'm dividing by that, and by that, so effectively
I'm dividing by both of them.
So there we've got them both written down.
Now delta x is a small positive quantity,
so I can divide top and bottom there by delta x, canceling delta x out.
So delta y over delta x is equal to -1 over, x + delta x, times by x.
Let me just multiply that denominator out so I
have x squared plus delta x times by x.
dy by dx is equal to the limit as delta x tends to 0 of
delta y over delta x.
Which will be the limit as delta x tends to 0
of -1 over x squared plus delta x times by x.
Equals... Let's have a look at what's happening in this denominator.
As delta x becomes very, very small x times by delta x is also very very small
and very much smaller than x squared.
So as delta x goes off to 0, this term here goes away to 0 and
just leaves us with the x squared.
So our limit is -1 over x squared.
And that's our calculation done in function notation to find the
derivative, as we call dy by dx, or the gradient of the curve at a point.
Now, the calculations that we've done or all quite complicated.
You need to do one or two to practice the method and
to see what's going on with it - to see how it's actually working.
But, to do most of your differentiation, there are a set of rules that you
can follow that will help you do the differentiation much more quickly.