Partial and Total
derivatives
Derivative of a function of several
variables
Notation and procedure
Partial and Total derivatives

Last week we saw:



That slope of a function f(x) is given by the first
derivative of that function f’(x)
We also saw the general rule that allows us to
find the derivative, as well as some particular
rules
This week we extend this to the case where
the function is a function of several
variables

As we saw in previous weeks, in economics we
often have functions of several variables
Partial and Total derivatives
Partial and total derivatives: concept
and notation
Rules for carrying out partial and total
differentiation
Concepts and notation


Imagine that we want to find the slope of a
function of x and y
Example
z  f  x, y 
z  2x  3xy  y  1
2

3
We know how to do this for a simple function of 1
variable

The concept of total and partial derivatives extends
the idea to functions of several variables
Concepts and notation
z  f  x, y 

The general idea is that we differentiate the
function twice, and get slopes in 2 directions

Once as if f were a function of x only

Once as if f were a function of y only

These 2 are the partial derivatives

The sum of the two terms is the total derivative
Concepts and notation
y
The lines give
us locations
with the same
“altitude” z
z=5
z = 10
z = 15
z = 20
z = 25
z = 30
x
Concepts and notation
y
slope in the y
direction
slope in the x
direction
x
Concepts and notation
z  f  x, y 

In order to be able to differentiate the
function twice (with respect to x and y), we
need to introduce some more notation


This is so that we don’t get confused when
differentiating
The notation separates the partials more
clearly that the f’ notation
Concepts and notation
z  f  x, y 

In fact, we re-use the notation from last
week, but slightly modified:
Slope 
f  x 
df  x 
x
dx
f  x 
x

for total derivatives
for partial derivatives
Another less used notation is:
f  x, y 
f  x, y 
 f y  x, y 
 f x  x, y 
y
x
Concepts and notation

For the case of a function of one variable, all
these notations are equivalent
f  x  df  x 

 f  x
x
dx

This is why there was no ‘notation problem’
last week

We didn’t need to separate the different
derivatives
Partial and Total derivatives
Partial and total derivatives: concept
and notation
Rules for carrying out partial and total
differentiation
Rules of partial/total differentiation

The rules of differentiation do not change
from last week


It is just the order in which things are done, and
the meaning of the partial derivative which is
different
The general rule, with a function of several
variables is:


Calculate the partial derivatives for each of the
variable, keeping the other variables constant
Add them up to get the total derivative
Rules of partial/total differentiation
f (x)
f (x)
Example
k (constant)
0
f(x) = 3  f’(x)=0
x
1
f(x) = 3x  f’(x)=3
n x n 1
1
2 x
f(x) = 5x²  f’(x)=10x
x
n
x

These stay the same.
Rules of partial/total differentiation

Let’s practise calculating partial derivatives:
f  x, y   2x2  3xy  y3 1
f  x, y 
 f x  x, y   4 x  3 y
x
f  x, y 
 f y  x, y   3x  3 y 2
y
y is treated like a constant
x is treated like a constant
Rules of partial/total differentiation

Given the partial derivatives, the total
derivative is obtained as follows:

The general definition of the total derivative is:
df  x, y  

f  x, y 
f  x, y 
dx 
dy
x
y
Let’s replace these to get the total derivative
df  x, y    4 x  3 y  dx   3x  3 y 2  dy
Rules of partial/total differentiation

In economics, we use partial derivatives most
of the time


This means that the main change from last
week is just one of notation, and of knowing
how to keep the other variables constant
Next week we shall see how we can use the set
of partial derivatives of a function of several
variables to find a maximum/minimum