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 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 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 x The lines give us locations with the same “altitude” z z = 5 z = 10 z = 15 z = 20 z = 25 z = 30

Concepts and notation y x slope in the x direction slope in the y direction

Concepts and notation 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 In fact, we re-use the notation from last week, but slightly modified: Another less used notation is: for total derivatives for partial derivatives

Concepts and notation 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

Partial and Total derivatives 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 x1 f(x) = 3x  f ’ (x)=3 f(x) = 5x ²  f ’ (x)=10x These stay the same.

Let’s practise calculating partial derivatives: Rules of partial/total differentiation y is treated like a constant 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 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 Rules of partial/total differentiation