2. Univariate Calculus
In this section, we consider the rate of change of a variable y
in response to a change in another variable x, where the two
variables are related to each other by the function
the variable y will represent the equilibrium value of an
endogenous variable, and x will be some parameter.
3. Cont…
• Rate of Change, the Difference Quotient and the
Derivative
• When the variable x changes from the value x0 to a new value x1, the
change is measured by the difference x1 – x0.
• Hence, using the symbol for "difference" to denote the change, we
write x = x1 – x0.
• Also needed is a way of denoting the value of the function f(x) at
various values of x.
• Thus, for the function f(x) =
4. Cont…
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
That is, is a function of x0 and .
7. Cont…
• A derivative is a function;
• in fact, in this usage the word derivative really means a derived
function.
• The original function y = f(x) is a primitive function, and the
derivative is another function derived from it.
• The derivative is merely a limit of the difference quotient, which
measures a rate of change of y, the derivative must of necessity also
be a measure of some rate of change.
• In view of the fact that the change in x envisaged in the derivative
concept is infinitesimal (i.e., ), however, the rate measured by the
derivative is in the nature of an instantaneous rate of change.
8. Cont…
• There is the matter of notation for a derivative
function.
• We may define the derivative of a given function
y = f(x) as follows:
For example, y= 3𝑥2-4
we have shown its difference quotient to be 6xo+3∆x, and the limit
of that quotient to be 6xo. Based on the latter, we may write (replacing xo with x):
9. Cont…
• we have drawn a total cost curve C, which is the graph of the
(primitive) function C = f(Q).
• Suppose that we consider as the initial output level from which an
increase in output is measured, then the relevant point on the cost
curve will be A.
• If output is to be raised to + ∆Q = Q2 , the total cost will be
increased from C0 to C0 + ∆C = C2; thus ∆C/∆Q = (C2 – C0)/(Q2 –
Q0).
• Geometrically, this is the ratio of two line segments, EB/AE, or the
slope of the line AB.
• This particular ratio measures an average rate of change – the
average marginal cost for the particular ∆Q pictured – and
represents a difference quotient.
• As such, it is a function of the initial value Q0 and the amount of
change ∆Q.
11. Cont…
• A Brief Review Rules of Differentiation
1. Constant-Function Rule
If y = k where k is a constant,
0
dx
dy
e.g. y = 10 then 0
dx
dy
12. Cont…
2. The Power Function Rule
If y = axn
, where a and n are constants
1
n
x
.
a
.
n
dx
dy
i) y = 4x => 4
4 0
x
dx
dy
ii) y = 4x2
=> x
dx
dy
8
iii) y = 4x-2
=>
3
8
x
dx
dy
13. Cont…
3. The Sum-Difference Rule
If y = f(x) g(x)
dx
)]
x
(
g
[
d
dx
)]
x
(
f
[
d
dx
dy
If y is the sum/difference of two or more
functions of x:
differentiate the 2 (or more) terms
separately, then add/subtract
(i) y = 2x2
+ 3x then 3
4
x
dx
dy
(ii) y = 5x + 4 then 5
dx
dy
14. Cont…
4. Product Rule
The derivative of the product of two (differentiable) functions is
equal to the first function times the derivative of the second plus the
second function times the derivative of the first:
As an example, consider the derivative of y= (2x+3)(3𝑥2
)
Let f(x) = 2x + 3 and g(x) = 3𝑥2
𝑑
𝑑𝑥
(2x + 3)(3𝑥2 ) = (2x+3)(6x) + (2)(3𝑥2 ) = 18𝑥2 + 18x
15. Cont…
• 4. Quotient Rule
• The derivative of the quotient of two functions, f(x)/g(x):
•
𝑑
𝑑𝑥
𝑓(𝑥)
𝑔(𝑥)
=
𝑓′ 𝑥 𝑔 𝑥 −𝑓(𝑥)𝑔′ 𝑥
𝑔2(𝑥)
2
2
4
2
4
1
2
1
4
4
2
x
x
x
x
dx
dy
x
x
y
Eg.
