Derivatives of a function of two variables

 Goal: Calculate the partial derivatives of a function of two variables

 Derivatives of a function of two variables

We have seen that some methods of the function of one variable can be applied to the function of two variables. In the notation of Leibniz dy/dx, x is the independent variable and y is the dependent variable. In a function of two variables, how do we adapt this notation? What would be an acceptable definition for the derivative of a function of two variables. This leads to the notion of partial derivatives

Definition

Let f be a function of two variables. Then the partial derivative of with respect to x, written as    

 is defined as:  

The partial derivative of f with respect to y, written as is defined as

 

N.B The notation   is called partial derivative of f with respect to x. 

The notation  is called partial derivative of with respect to y.

Example 

Use the definition of the partial derivative as a limit to calculate  and

for the function :

 

Solution

First, let's calculate f(x+h, y):

  

Let's substitute this in the definition of the partial derivative of f with respect to x.

 

 

To calculate the partial derivative of f with respect to y, let's calculate f(x, y+h)

 

Let's substitute this in the definition of the partial derivative with respect to y:

 

Practice

Use the definition of the partial derivative as a limit to calculate and for the following function: