For a (smooth) function f(x,y) of two variables, we may take partial derivatives with respect to either variable. The first time we do this, we get the (first order) partial derivative functions fx(x,y) and fy(x,y), or, more briefly, simply
fx, fy.If we take the partial derivatives of these two functions, (and assuming sufficient smoothness that we can interchange orders of differentiation) we will come up with three second-order partial derivative functions:
fxx, fxy, fyy.And if we differentiate k times, then there are k+1 different functions that can appear in the answer, although exactly which ones occur depends on how many times we differentiate with respect to x and how many times with respect to y.
Now suppose that the number of independent variables is n. (For the above problem, n was 2). It is easy to see that the number of first order partial derivatives is n, but can you determine a formula for the number of distinct partial derivative functions of order k?
This puzzle courtesy of Rich Fabiano, via Fritz Keinert.
I give up, show me the solution.