Notice, in the example above, the two mixed partial derivatives ∂x∂y∂2f and ∂y∂x∂2f are the same. This is not a coincidence; it happens for almost every function you encounter in practice. For example, look at what happens to a general polynomial term xnyk:
Technically, the symmetry of second derivatives is not always true. There is a theorem, referred to variously as Schwarz's theorem or Clairaut's theorem, which states that symmetry of second derivatives will always hold at a point if the second partial derivatives are continuous around that point. To really get into the meat of this, we'd need some real analysis.
You should keep in the back of your mind that exceptions exist, but the symmetry of second derivatives work for just about every "normal" looking function that you will come across.
These mixed derivatives ∂x∂y∂2f and ∂y∂x∂2f evaluated at the origin (0,0) turn out to be 1 and −1 respectively. Computing this is actually pretty tricky, and requires looking directly at the limit-based definition of the derivative. Wikipedia provides a nice explanation, should you find yourself feeling ambitious.
Example 2: Higher order derivatives
Why stop at second partial derivatives? We could also take, say, five partial derivatives with respect to various input variables.
Problem: If f(x,y,z)=sin(xy)ex+z, what is fzyzyx?
Solution: The notation fzyzyx is shorthand for ((((fz)y)z)y)x, so we differentiate with respect to z, then with respect to y, then z, then y, then x. That is, we read left to right.
It's worth pointing out that the order is different in the other notation:
So the order of differentiation is indicated by the order of the terms in the denominator from right to left.
Anyway, back to the problem at hand. This is one of those tasks where you just have to roll up your sleeves and slog through it, but to help things let's color the variables x,y,z to keep track of where they all are:
Man! That was a tedious example. But if you could follow all the way through, computing multiple partial derivatives should not be an issue for you. It's one of those things that just requires more bookkeeping than anything else.