partial and total derivatives derivative of a function of several variables notation and procedure

Partial and Total derivatives

Derivative of a function of several variables

Notation and procedure


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: 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

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

,z f x y

2 32 3 1z x xy 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

,z f x y


y

x

The lines give us locations with the same “altitude” z

z = 5

z = 10

z = 15

z = 20

z = 25

z = 30


y

x

slope in the x direction

slope in the y direction


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

,z f x y


In fact, we re-use the notation from last week, but slightly modified:

Another less used notation is:

,z f x y

f x

Slopex

df x

dx

f x

x

for total derivatives

for partial derivatives

,,x

f x yf x y

x

,,y

f x yf x y

y


For the case of a function of one variable, all these notations are equivalent

This is why there was no ‘notation problem’ last week We didn’t need to separate the different

derivatives

f x df xf x

x dx


Partial and total derivatives: concept and notation

Rules for carrying out partial and 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


Example

k (constant) 0 f(x) = 3 f’(x)=0

x 1 f(x) = 3x f’(x)=3

f(x) = 5x² f’(x)=10x

f (x)

nx

x

f (x)

n 1n x

1

2 x

These stay the same.

Let’s practise calculating partial derivatives:


2 3, 2 3 1f x y x xy y

,, 4 3x

f x yf x y x y

x

y is treated like a constant

2,, 3 3y

f x yf x y x y

y

x is treated like a constant

Given the partial derivatives, the total derivative is obtained as follows: The general definition of the total derivative

is:

Let’s replace these to get the total derivative


, ,,

f x y f x ydf x y dx dy

x y

2, 4 3 3 3df x y x y dx x y dy

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


partial and total derivatives derivative of a function of several variables notation and procedure

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