Calculating mixed partial derivatives of a function A(x,y)

I have a script in MATLAB to calculate the value of a function A. I have also programmed the analytic derivatives of A with respect to x and y and also d2A/dxdy. Now, I want to check that my analytic derivatives are correct with a numerical validation. The form of the function is very complicated, so it's not so easy as just re-checking my math. How would you program the numerical derivative of a mixed partial derivative in MATLAB?

Answer 1

[Roger Stafford]

   syms x y
   f(x,y) = x*sin(y);
   d2fdxdy = diff(diff(f,x),y); % = cos(y)

Another way is:

   syms x y
   diff(x*sin(x*y),x,y);  % = 2*x*cos(x*y)-x^2*y*sin(x*y)

[Caleb]

So diff actually produces analytic derivatives, correct? Can it do numerical?

[Caleb]

Additionally, if my function contains expressions that are functions of x and y, such as: A(x,y) = x*cos(y) + y*C(x,y), how is that handled?

[Roger Stafford]

@Caleb: Obviously you cannot find derivatives of such a function without having an explicit expression for C(x,y).

   C(x,y) = ..... % <-- whatever you define C as in terms of x and y
   A(x,y) = x*cos(y) + y*C(x,y);
   d2Adxdy = diff(A,x,y);

See:

   http://www.mathworks.com/help/symbolic/diff.html