# Partial derivatives with respect to multiple variables

For any natural number $N$:

$$_t k_{mn}^1 = \frac{1}{m!n!}\frac{\partial^{n+m} k_1 (0, 0)}{\partial x^m \partial t^n}, \qquad (m,n=0,1,\ldots, N)$$

where $k_1(x,t)$ is a known function, for example $k_1(x,t) = x^5 + t^3$.

How I can write a code in Mathematica to calculate these derivatives?

• @march There is no help about the higher order derivatives of functions of more than one variable in Mathematica! How I can do that? Jul 25, 2015 at 7:24
• @NoMan That is not true. See here (reference.wolfram.com/language/ref/D.html), third entry at the top. And see my answer on how to use it Jul 25, 2015 at 8:39

I suspect that you want to evaluate the derivatives at $(0,0)$ after performing the derivative. Otherwise the whole expression would vanish anyways.
In the documnetnation for D you will find that there is indeed a way to achieve what you want: D[f[x1,...,xn],{x1,a1},...,{xn,an}] successively computed the $a_i$th derivative of f with respect to its variables.
k1[x_, t_] := x^5 + t^3;

which will give you zero for this particular choice of k1.