Multiplicative constant: if y = kf (x) then
dy
= kf 0 (x)
dx
Derivatives in Economics
This leaflet is an overview of differentiation and its applications in Economics.
The marginal cost M C is the rate of change of the total
dy
0
0
= f (x) ± g (x) cost function T C: M C = dT C , where q is the output.
Addition rule: if y = f (x) ± g(x) then
dx
dq
Similarly, the marginal revenue M R is the rate of change
Product rule: if y = f (x) × g(x) then
dT R
of the total revenue function T R: M R =
. When
dy
0
0
dq
= f (x)g(x) + f (x)g (x)
M R is positive, T R is an increasing function of q, and
dx
when M R is negative, T R is a decreasing function of q.
f (x)
Quotient rule: if y =
then
The elasticity E of a function q = f (p) is the rate of prog(x)
portionate change in q given a proportionate change in p:
dq
0
0
dy
f (x)g(x) − f (x)g (x)
d ln q
q
=
E = dp =
. This is the slope of the function when
dx
(g(x))2
d
ln p
p
plotted on a log-log scale.
Chain rule (derivative of a function of a function):
Author: Morgiane Richard, University of Aberdeen
if y = g(u) with u = f (x) then
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Differentiation for Economics and Business Studies
Functions of one variable
Reviewer: Anthony Cronin, University College Dublin
The derivative of a function f is a new function obtained
df
by differentiating f . It can be written f 0 or
. It is the
dx
rate of change of f and gives information on the shape
and optimum values of f .
Table of Derivatives
y = f (x)
k constant
x
x2
xn
ex
ln x
eax+b
ln (ax + b)
ln (f (x))
dy
= f 0 (x)
dx
0
1
2x
nxn−1
ex
1
x
aeax+b
a
ax0 + b
f (x)
f (x)
Rules of Differentiation
For any function f and g and any constant value k:
Additive constant: if y = f (x) + k then
dy
= f 0 (x)
dx
2
dy
dy du
=
= g 0 (u) × f 0 (x)
dx
du dx
Shape of Function
sign of
dy
= f 0 (x)
dx
>0
>0
<0
<0
sign of
d2 y
= f 00 (x)
dx2
>0
<0
>0
<0
1.5
concave decreasing function
1
shape of the curve of f
0.5
convex decreasing
function
increasing and convex
increasing and concave
decreasing and convex
decreasing and concave
Stationary points
0
0
0.5
1
1.5
2
Figure 1: Examples of decreasing concave and convex functions
4
First Order Condition (FOC): if a point x0 is such that
f 0 (x0 ) = 0, then it is a stationary point. It can be a maximum, or a minimum, or an inflection point.
Second Order Condition (SOC): the sign of the second
derivative indicates whether the optimum is a maximum,
minimum or inflection point:
3.5
convex increasing function
3
2.5
2
1.5
concave increasing function
1
0.5
value of
dy
(x0 ) = f 0 (x0 )
dx
0
0
0
sign of
d2 y
(x0 ) = f 00 (x0 )
dx2
>0
<0
0
Nature of
point at x0
minimum
maximum
inflection
0
0
1
2
3
5
Figure 2: Examples of increasing concave and convex functions
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