Derivative of a Continued Fraction Function

MAIN Say we have a continued fraction function f(x) = {1+{1+{1+{1+\cdots}}}} = {1+} and we wanted to find the derivative without knowing the right hand side. Then if we take the partial convergents to the function f(x)_1 = {1}\\ f(x)_2 = {1+{1}}\\ f(x)_3 = {1+{1+{1}}}\\ f(x)_4 = {1+{1+{1+{1}}}}\\ and take their derivatives (as series expansions). We find that the series expansions all tend to the same series f(x) = 1-2x+6x^2-20x^3+70x^4-252x^5+924x^6-\cdots where the coefficients are the signed central binomial coefficients A000984 a(n) = (-1)^n {n!^2}, \; n=0,1,2,\cdots we have that ^\infty (-1)^n {n!^2} x^n = {} which agrees with the original right hand side of equation 1. This may be useful to work out the closed forms for some continued fractions. If we take the derivative until a known integer sequence appears, we could potentially integrate back to the closed form.