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A general approach to rates of change
Using a similar approach to that in the previous slide show it is possible to find the gradient at any point (x, y) on the curve y = f(x).
At this point it is useful to introduce some “new” notation due to Leibniz.
The Greek letter ∆(delta) is used as an abbreviation for “the increase in”.
Thus the “increase in x” is written as ∆x, and the “increase in y” is written as ∆y.
P (x1, y1)
Q (x2, y2)
x
y
O
12 yy
12 xx
So when considering the gradient of a straight line, ∆x is the same as x2 – x1 and ∆y is the same as y2 – y1.
∆x
∆y
Gottfried Leibniz1646 - 1716
gradient
Note∆x is the same as h used on the previous slide show.
Gradient of the curve y = x2 at the point P(x, y)Suppose that the point Q(x + ∆x, y + ∆y) is very close to the point P on the curve.
The small change from P in the value of x is ∆x and the corresponding small change in the value of y is ∆y.
It is important to understand that ∆x is read as “delta x” and is a single symbol.
The gradient of the chord PQ is:
xO
y
P(x, y)
Q(x + ∆x, y + ∆y)
y = x2
x
y y∆x
∆y
The coordinates of P can also be written as (x, x2) and the coordinates of Q as [(x + ∆x), (x + ∆x)2].
So the gradient of the chord PQ can be written as:
= 2x + ∆x
So = 2x + ∆x
As ∆x gets smaller approaches a limit and we start to refer to it in theoretical terms.
This limit is the gradient of the tangent at P which is the gradient of the curve at P.
It is called the rate of change of y with respect to x at the point P.
x
yxy
x δ
δdd
0δlim
For the curve y = x2,
xxxy
xδ2
dd
0δlim
= 2x
This is the result we obtained previously.
This is denoted by or .xy
dd
Gradient of the curve y = f(x) at the point P(x, y)
xO
y
P(x, y)
Q(x + δx, y + δy)
y = f(x)
x
y yδx
δy
For any function y = f(x) the gradient of the chord PQ is:
xy
xxxyyy
δδ
)δ()δ(
The coordinates of P can also be written as (x, f(x)) and the coordinates of Q as [(x + δx), f(x + δx)].
So the gradient of the chord PQ can be written as:
xxxxxx
)δ(
)(f)δ(f
x
yxy
x δ
δdd
0δlim
xxx
xxx
x )δ(
)(f)δ(f
0lim
x
xxx
x δ
)(f)δ(f
0lim
It is defined by
In words we say:
The symbol is called the derivative or the differential coefficient of y with respect to x.xy
dd
x
xxxxy
x δ
)(f)δ(fdd
0δlim
“dee y by dee x is the limit of as δx tends to zero”xy
δδ
“tends to” is another way of saying “approaches”
DEFINITION OF THE DERIVATIVE OF A FUNCTIONDEFINITION OF THE DERIVATIVE OF A FUNCTION
Sometimes we write h instead of ∆x and so the derivative of f(x) can be written as
h
xhxxy
h
)(f)(fdd
0lim
DEFINITION OF THE DERIVATIVE OF A FUNCTIONDEFINITION OF THE DERIVATIVE OF A FUNCTION
If y = f(x) we can also use the notation:
xy
dd = f’ (x)
In this case f’ is often called the derived function of f. This is also called f-prime.
The procedure used to find from y is called differentiating y with respect to x.xy
dd
Find for the function y = x3.
Example (1)
In this case, f(x) = x3.
xy
dd
h
xhxxy
h
)(f)(fdd
0lim
h
xhx
h
33
0
)(lim
h
xhxhhxx
h
33223
0
33lim
h
hxhhx
h
322
0
33lim
22
033lim hxhx
h
= 3x2
f(x) f ‘ (x)
x2 2x
x3 3x2
Results so far
−4 −3 −2 −1 1 2 3 4
5
10
15
x
yy = x3
−4 −3 −2 −1 1 2 3 4
5
10
15
x
yy = x3
Gradient at (2, 8) = 12
Find for the function y = x4.
Example (2)
In this case, f(x) = x4.
xy
dd
h
xhxxy
h
)(f)(fdd
0lim
h
xhx
h
44
0
)(lim
h
xhxhhxhxx
h
4432234
0
464lim
3223
0464lim hxhhxx
h
= 4x3
Results so far
h
hxhhxhx
h
43223
0
464lim
f(x) f ‘(x)
x2 2x
x3 3x2
x4 4x3
Find for the function y = .
Example (3)
xy
dd
x1
In this case, f(x) = .x1
h
xhxxy
h
)(f)(fdd
0lim
h
xhxh
11
0lim
h
hxxhxx
h
)()(
0lim
h
hxxh
h
)(
0lim
)(0lim
hxxh
h
h
)(
1
0lim
hxxh
21x
So we now have the following results:
We also know that if y = x = x1 then )1or ( 1dd 0x
xy
and if y = 1 = x0 then )0or ( 0dd 1 x
xy
These results suggest that if y = xn then 1dd nnx
xy
It can be proven that this statement is true for all values of n.
n xn
–1
2 x2 2x
3 x3 3x2
4 x4 4x3
xy
dd
)or ( 1 1xx)or ( 1 2
2 x
x
