Building expression trees is not the only way to do automatic differentiation. A simpler way is to just carry the value and it's derivative as a pair:

    #include <math.h>
    #include <stdio.h>

    struct autodiff { double value, deriv; };

    autodiff just(double x) { return { x, 1 }; }

    autodiff operator +(autodiff a, autodiff b) {
        return { a.value + b.value, a.deriv + b.deriv };
    }

    autodiff operator *(autodiff a, autodiff b) {
        return { a.value * b.value, a.deriv*b.value + a.value*b.deriv };
    }

    autodiff sin(autodiff a) {
        return { sin(a.value), cos(a.value)*a.deriv };
    }

    int main() {
        autodiff x = just(.1);
        for (int ii = 0; ii<4; ii++) {
            x = x*x + sin(x);
        }
        printf("value: %lf, deriv: %lf\n", x.value, x.deriv);
        return 0;
    } 

There is no need to differentiate the for loop or the semicolons. This way is not doing symbolic differentiation. It's implementing the differentiation rules in parallel to calculating the values at run time.

This generalizes to partial derivatives for multivariate functions too:

     template<int dims>
     struct autograd { double value, grad[dims}; };