Simple term rewriting system (TRS) that is based on structural pattern matching, see https://peps.python.org/pep-0636 and https://inst.eecs.berkeley.edu/~cs294-260/sp24/2024-01-22-term-rewriting

trs.py

def topdown(rule, expr):
    match rule(expr):
        case (spec, *args):
            return (spec, *(topdown(rule, arg) for arg in args))
        case expr:
            return expr

def rewrite(rule, expr):
    while (new_expr := topdown(rule, expr)) != expr:
        expr = new_expr
    return new_expr

def derive(expr):
    match expr:
        case ('∂', ('add', u, v), var):
            return ('add', ('∂', u, var), ('∂', v, var))
        case ('∂', ('pow', var0, int(n)), var) if var == var0:
            return ('mul', n, ('pow', var, n - 1))
        case ('∂', ('div', 1, var0), var) if var == var0:
            return ('∂', ('pow', var, -1), var)
        case expr:
            return expr

def simplify(expr):
    match expr:
        case ('add', a, ('unm', b)) | ('add', ('unm', b), a):
            return ('sub', a, b)
        case ('mul', -1, var) | ('mul', var, -1):
            return ('unm', var)
        case ('pow', var, 1):
            return var
        case expr:
            return expr

OPS = {
    'add': '{%s}+{%s}',
    'sub': '{%s}-{%s}',
    'div': '\\frac{%s}{%s}',
    'mul': '{%s}{%s}',
    'pow': '{%s}^{%s}',
    '∂': '\\frac{\\partial{\\left(%s\\right)}}{\\partial{%s}}',
}

def tex(tree):
    match tree:
        case (spec, *args):
            return OPS[spec] % tuple(map(tex, args))
        case atom:
            return atom

expr = ('∂', ('add', ('pow', 'x', 2), ('div', 1, 'x')), 'x')
derv = rewrite(derive, expr)
simp = rewrite(simplify, derv)
print(expr)  # Original expression.
print(derv)  # Derivative of the expression.
print(simp)  # Simplified derivative of the expression.
print(tex(expr), '=', tex(simp))  # LaTeX markup.