Functions of Several Variables

  p3=Plot3D[Sin[x y],{x,0,3},{y,0,3}];

  Show[p3,ViewPoint->{0,-2,0}];

  Show[p3,Shading->False];

Partial derivatives hold one or more of the variables constant and differentiate with respect to the variable that is not fixed.

Here are some first partial derivatives.

  D[x^2 y + 2 Sin[y],x]

  D[x^2 y + 2 Sin[y],y]

Here are some second partial derivatives.

First Dxy, that is, partial with respect to x first, then with
respect to y.

  D[x^2 y + 2 Sin[y],y,x]

Next Dyx

  D[x^2 y + 2 Sin[y],x,y]

Now, D^3 y, the third partial with respect to y.

  D[x^2 y + 2 Sin[y],{y,3}]

You will recall that for "nice" functions, that is, functions whose first and second partials, along with f itself, are continuous, we always have the mixed second partials being equal.

  Clear[x,y,z,g,h]
  g[x_,y_]:=Sin[x^2 y]
  h[x_,y_,z_]:=x Exp[y] Sin[Pi z]

The following (double equal signs, ==) is Mathematica's way of asking if something is true or false.

  D[g[x,y],x,y]==D[g[x,y],y,x]

  D[h[x,y,z],x,y]==D[h[x,y,z],y,x]

  D[h[x,y,z],x,z]==D[h[x,y,z],z,x]

  D[h[x,y,z],z,y]==D[h[x,y,z],y,z]

Up to Vector Differential Calculus