Supposing we have a function like
$$\prod_{n=0}^{5} \frac{(x+n)}{(1+n)}$$
and want to obtain the derivative at x = 0
Is it valid to convert to a polynomial form
$$\ \frac{x^6}{720}+\frac{x^5}{48}+\frac{17 x^4}{144}+\frac{5 x^3}{16}+\frac{137 x^2}{360}+\frac{x}{6} $$
and then obtain the derivative
$$\ \frac{x^5}{120}+\frac{5 x^4}{48}+\frac{17 x^3}{36}+\frac{15 x^2}{16}+\frac{137 x}{180}+\frac{1}{6} $$
so that we can substitute x = 0 and the result is 1/6 ?