logic.py

            return u * diff(v, x) + v * diff(u, x)
        elif op == '/':
            return (v * diff(u, x) - u * diff(v, x)) / (v * v)
        elif op == '**' and isnumber(x.op):
            return (v * u ** (v - 1) * diff(u, x))
        elif op == '**':
            return (v * u ** (v - 1) * diff(u, x) +
                    u ** v * Expr('log')(u) * diff(v, x))
        elif op == 'log':
            return diff(u, x) / u
        else:
            raise ValueError("Unknown op: {} in diff({}, {})".format(op, y, x))


def simp(x):
    """Simplify the expression x."""
    if isnumber(x) or not x.args:
        return x
    args = list(map(simp, x.args))
    u, op, v = args[0], x.op, args[-1]
    if op == '+':
        if v == 0:
            return u
        if u == 0:
            return v
        if u == v:
            return 2 * u
        if u == -v or v == -u:
            return 0
    elif op == '-' and len(args) == 1:
        if u.op == '-' and len(u.args) == 1:
            return u.args[0]  # --y ==> y
    elif op == '-':
        if v == 0:
            return u
        if u == 0:
            return -v
        if u == v:
            return 0
        if u == -v or v == -u:
            return 0
    elif op == '*':
        if u == 0 or v == 0:
            return 0
        if u == 1:
            return v
        if v == 1:
            return u
        if u == v:
            return u ** 2
    elif op == '/':
        if u == 0:
            return 0
        if v == 0:
            return Expr('Undefined')
        if u == v:
            return 1
        if u == -v or v == -u:
            return 0
    elif op == '**':
        if u == 0:
            return 0
        if v == 0:
            return 1
        if u == 1:
            return 1
        if v == 1:
            return u
    elif op == 'log':
        if u == 1:
            return 0
    else:
        raise ValueError("Unknown op: " + op)
    # If we fall through to here, we can not simplify further
    return Expr(op, *args)


def d(y, x):
    """Differentiate and then simplify."""
    return simp(diff(y, x))