Calculating partial derivatives of a function of two variables without using limit

 Goal: Calculate partial derivatives without using limit

The idea behind calculating partial derivatives is to keep all independent variables, other than the ones with respect to which you are differentiating, as constant. Then proceed as if you were using a function of single variable. To prove this let's fix y and pose g(x) = f(x, y)

The same is true when differentiating with respect to y. In this case, we fix x and pose  h(y) = f(x,y) 

as a function of y.                                                                                                                                        

                                                                                                                                               

Example

Calculate 

for the following functions by holding the opposite variable constant and differentiating

Solution

The derivative of the third, fifth and sixth terms are all zero because they do not contain the variable x. Therefore they are treated as constants. The derivative of the second term is -3y since y is considered as a constant number.

Then differentiate g(x,y) with respect to x using 

    
the chain rule and the power rule.

Then differentiate g(x,y) with respect to y using

 the chain rule and the power rule.

   

Practice

by holding the opposite variable 

constant and then differentiating