For the function given above, you can use the usual rules of differentiation to find the partial derivatives of at points where neither x nor y is zero. But if you want to find for example (where y≠0) then you have to evaluate the limit given by the definition . To find this limit, you'll need to use the fact that . Leaving out the details, this gives .
To find , you again have to go back to the definition of the derivative, but this time it's easier: . So in fact the formula still works when y=0.
To calculate the second partial derivative you again have to go back to the definition: .
For the other mixed partial derivative , *if it exists*, you need to use the same procedure, but this time finding and first, and then using .