The rules of symbolic differentiation, such as,

d          d       d
- (A+B) =  - A  +  - B
dx         dx      dx

can be easily expressed in Prolog, e.g.,

diff(plus(A,B), X, plus(DA, DB))
   <= diff(A, X, DA) and diff(B, X, DB).

NB. The 'and not equal(Y, X)' in the definition of 'diff(.,.,.)' above is necessary to prevent d/dx 7=1, say.

It is a good exercise, and not so easy, to write Prolog to simplify the result of differentiation, for example, to replace times(1,x)) with x and so on.

Just in case you were wondering, the differentiator cannot be run backwards to get an integrator, which is a pity. It is possible to write a program to perform symbolic integration in Prolog, but it is far from easy (in any programming language) if more than polynomials must be handled.

There are more Prolog examples here.