Partial derivatives of a function of three variables

 The partial derivative of a function of three variables can be calculated the same way we calculate the partial derivatives of a function of two variables. To calculate the partial derivative with respect to x, we consider y and z constants then we calculate the derivative considering only x as a variable. In general to calculate the partial derivative to one variable, we calculate the derivative with respect the considered variable while we keep the other two variables constant.

Definition

Example

Solution

To  calculate the partial derivative with respect to x using the limit definition, let's start by using the formula                                                                                                                                              

Let's start by calculating f(x + h, y, z):                                                                                                         

Let's substitute f(x+h, y, z) and f(x,y,z) in the formula:                                                                                   

                                                                                                     

Practice