LESSON 4. DERIVATIVES
Mathematics – 2011/201 2- 1 -
LESSON 4
DERIVATIVES
1. Rate of change and the derivative.
Let us suppose that we are interested in the rate of change of any variable y in response
to a change in another variable x, where the two variables are related by the function,
i.e. y=f(x).
The variable y could be for example, the equilibrium value of an endogenous
variable, and x a parameter.
The difference quotient:
When a variable x changes from x
0
to x
1
, the change us measured by the
difference x
1
-x
0
. Let denote this change by
01
xxx −=∆
.
When x changes from an initial value x
0
to a new variable
xxx ∆+=
01
the value of the
function y=f(x) changes from f(x
0
) to
)(
0
xxf ∆+
. The change in y per unit of change in x
can be defined by the difference quotient.
x
xfxxf
x
y
∆
−∆+
=
∆
∆)()(
00
This quotient which measures the average rate of change of y, can be calculated if we
know the initial value of x, or x0 and the magnitude of change in x, or Dx.
Example: Given y=f(x)=3x
2
-4, we can write
4)(3)(4)(3)(
2
00
2
00
−∆+=∆+−= xxxxfxxf
.
Therefore:
xx
x
xxx
x
xxx
x
y∆+=
∆
∆+∆
=
∆
−−−∆+
=
∆
∆36
)(36)43(4)(3
0
2
0
2
0
2
0
If we know x0 and Dx, we can evaluate this quotient difference.
Definition
Frequently, we interested in the rate change of y when Dx is very small. In this case, it
possible to obtain an approximation of Dy/Dx by dropping all the terms in the difference
quotient involving the expression Dx.
If Dx approaches zero in the above example, we can write:
00
00
6)36(
limlim
xxx
x
y
xx
=∆+=
∆
∆
→∆→∆
