# The Mixed Partial Derivatives are NOT Always Equal

 Here is an example where the order in which we take the partial derivatives makes a big difference. Consider the following function of two variables:

> f := proc(x,y)
> if x = 0 and y = 0 then RETURN( 0 )
> else RETURN( x*y*(x^2-y^2)/(x^2+y^2) )
> fi;
> end;

```f := proc(x, y)
if x = 0 and y = 0 then RETURN(0)
else RETURN(x*y*(x^2 - y^2)/(x^2 + y^2))
fi
end```
> assume(a>0, b >0):
> f(a,b);
```                                        2     2
a~ b~ (a~  - b~ )
-----------------
2     2
a~  + b~```
> fx := unapply(diff(f(x,y),x),x,y);
```                              2    2        2          2     2    2
y (x  - y )      x  y       x  y (x  - y )
fx := (x, y) -> ----------- + 2 ------- - 2 --------------
2    2        2    2         2    2 2
x  + y        x  + y        (x  + y )```
> fx(0,y);
`                                      -y`
 That is the partial derivative of f w.r.t. x evaluated at (0,y) is -y for all values of y. Thus, fyx(0,0) = -1. On the other hand,

> fy := unapply(diff(f(x,y),y),x,y);

```                              2    2          2          2   2    2
x (x  - y )      x y        x y  (x  - y )
fy := (x, y) -> ----------- - 2 ------- - 2 --------------
2    2        2    2         2    2 2
x  + y        x  + y        (x  + y )```
> fy(x,0);
`                                       x`
 So the partial derivative of f w.r.t. y at (x,0) is x for all values of x. Hence, fxy(0,0) = 1. The two mixed partial derivatives are different at the point (0,0).

Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>