I want to algorithmically find derivatives and partial derivatives of expressions written in function notation. For example, I can type "d/dx f(x, y(x), g(x) * h(x))" directly into Wolfram Alpha, without defining any of the referenced functions more specifically. I'd like to work with code which is able to do this.
Unfortunately, I don't own Mathematica, but I do have basic MATLAB and anything for Windows that's free (Python, R, etc.). If the code is significantly easier to work with in Mathematica than anywhere else, I might get it, but free alternatives would be preferred.
How can I perform these sorts of calculations in code?