FIRST YEAR CALCULUS
W W L CHEN
c
±
W W L Chen, 1982, 2008.
This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990.
It is available free to all individuals, on the understanding that it is not to be used for ﬁnancial gain,
and may be downloaded and/or photocopied, with or without permission from the author.
However, this document may not be kept on any information storage and retrieval system without permission
from the author, unless such system is not accessible to any individuals other than its owners.
Chapter 3
INTRODUCTION TO DERIVATIVES
3.1. Introduction
We begin by looking at a simple example. Suppose that a car is travelling along a road. For 10 hours,
its distance from the point of origin is noted at hourly intervals and recorded. The table below shows
its distance
x
in kilometres from the point of origin against time
t
in hours:
t
0
1
2
3
4
5
6
7
8
9
10
x
0 50 120 190 290 350 470 560 620 690 750
We can denote by
s
(
t
) the distance of the car from the point of origin after time
t
, so that
s
(3) = 190
and
s
(8) = 620, for example. Then the average speed of the car between the 3hour mark and the 8hour
mark will be given by
change in distance over the time interval
length of the time interval
=
s
(8)

s
(3)
8

3
=
620

190
8

3
= 86
kilometres per hour. Suppose next that we wish to ﬁnd the actual speed of the car at the 3hour mark.
Then the table above is not of much use. However, if more precise information of the position of the car
is available at all time, then perhaps the following strategy may be useful. We take the position
s
(3) of
the car at the 3hour mark. Now add a small time interval Δ
t
, and ﬁnd out the position
s
(3 + Δ
t
) of
the car after 3 + Δ
t
hours. Then we calculate the average speed
s
(3 + Δ
t
)

s
(3)
Δ
t
of the car over this small time interval. If Δ
t
is very small, then this average should be roughly the
speed of the car at the 3hour mark. We are therefore looking at some quantity, if it exists at all, like
lim
Δ
t
→
0
s
(3 + Δ
t
)

s
(3)
Δ
t
.
Chapter 3 : Introduction to Derivatives
page 1 of 20
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View Full DocumentFirst Year Calculus
c
±
W W L Chen, 1982, 2008
Consider the graph of a function
y
=
f
(
x
). Suppose that
P
(
a,b
) is a point on the curve
y
=
f
(
x
).
Consider now another point
Q
(
x,y
) on the curve close to the point
P
(
). We draw the line joining
the points
P
(
) and
Q
(
), and obtain the picture below.
x
yy
=
f
(
x
)
P
(
a, b
)
Q
(
x, y
)
Clearly the slope of this line is equal to
y

b
x

a
=
f
(
x
)

f
(
a
)
x

a
.
Now let us keep the point
P
(
) ﬁxed, and move the point
Q
(
) along the curve towards the point
P
.
Eventually the line
PQ
becomes the tangent to the curve
y
=
f
(
x
) at the point
P
(
), as shown in the
picture below.
x
=
f
(
x
)
P
(
a, b
)
We are interested in the slope of this tangent line. Its value is called the derivative of the function
y
=
f
(
x
) at the point
x
=
a
, and denoted by
d
y
d
x
±
±
±
±
x
=
a
or
f
0
(
a
)
.
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 Calculus, Derivative, W W L Chen

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