2
Instantaneous Rate of Change:
The Derivative
Suppose that
y
is a function of
x
, say
y
=
f
(
x
). It is often necessary to know how sensitive
the value of
y
is to small changes in
x
.
EXAMPLE 2.1
Take, for example,
y
=
f
(
x
) =
radicalbig
625

x
2
(the upper semicircle of
radius 25 centered at the origin). When
x
= 7, we find that
y
=
√
625

49 = 24. Suppose
we want to know how much
y
changes when
x
increases a little, say to 7.1 or 7.01.
In the case of a straight line
y
=
mx
+
b
, the slope
m
= Δ
y/
Δ
x
measures the change in
y
per unit change in
x
. This can be interpreted as a measure of “sensitivity”; for example,
if
y
= 100
x
+ 5, a small change in
x
corresponds to a change one hundred times as large
in
y
, so
y
is quite sensitive to changes in
x
.
Let us look at the same ratio Δ
y/
Δ
x
for our function
y
=
f
(
x
) =
radicalbig
625

x
2
when
x
changes from 7 to 7
.
1. Here Δ
x
= 7
.
1

7 = 0
.
1 is the change in
x
, and
Δ
y
=
f
(
x
+ Δ
x
)

f
(
x
) =
f
(7
.
1)

f
(7)
=
radicalbig
625

7
.
1
2

radicalbig
625

7
2
≈
23
.
9706

24 =

0
.
0294
.
Thus, Δ
y/
Δ
x
≈ 
0
.
0294
/
0
.
1 =

0
.
294.
This means that
y
changes by less than one
third the change in
x
, so apparently
y
is not very sensitive to changes in
x
at
x
= 7.
We say “apparently” here because we don’t really know what happens between 7 and 7
.
1.
Perhaps
y
changes dramatically as
x
runs through the values from 7 to 7
.
1, but at 7
.
1
y
just happens to be close to its value at 7. This is not in fact the case for this particular
function, but we don’t yet know why.
17
18
Chapter 2 Instantaneous Rate of Change: The Derivative
One way to interpret the above calculation is by reference to a line. We have computed
the slope of the line through (7
,
24) and (7
.
1
,
23
.
9706), called a
chord
of the circle.
In
general, if we draw the chord from the point (7
,
24) to a nearby point on the semicircle
(7 + Δ
x, f
(7 + Δ
x
)), the slope of this chord is the socalled
difference quotient
slope of chord =
f
(7 + Δ
x
)

f
(7)
Δ
x
=
radicalbig
625

(7 + Δ
x
)
2

24
Δ
x
.
For example, if
x
changes only from 7 to 7.01, then the difference quotient (slope of the
chord) is approximately equal to (23
.
997081

24)
/
0
.
01 =

0
.
2919. This is slightly less
steep than the chord from (7
,
24) to (7
.
1
,
23
.
9706).
As the second value 7 + Δ
x
moves in towards 7, the chord joining (7
, f
(7)) to (7 +
Δ
x, f
(7 + Δ
x
)) shifts slightly. As indicated in figure 2.1, as Δ
x
gets smaller and smaller,
the chord joining (7
,
24) to (7 + Δ
x, f
(7 + Δ
x
)) gets closer and closer to the
tangent line
to the circle at the point (7
,
24). (Recall that the tangent line is the line that just grazes
the circle at that point, i.e., it doesn’t meet the circle at any second point.) Thus, as Δ
x
gets smaller and smaller, the slope Δ
y/
Δ
x
of the chord gets closer and closer to the slope
of the tangent line. This is actually quite difficult to see when Δ
x
is small, because of the
scale of the graph. The values of Δ
x
used for the figure are 1, 5, 10 and 15, not really very
small values. The tangent line is the one that is uppermost at the right hand endpoint.