Solutions

Our conjecture is that $S_n'(x)=nx^{n-1}$. Indeed,

$$\begin{align*} S_n'(x)&=\lim_{h\to 0}\frac{S_n(x+h)-S_n(x)}{h} \\ &=\lim_{h\to 0}\frac{(x+h)^n-x^n}{h} \\ &=\lim_{h\to 0}\frac{\sum_{i=0}^n {n\choose i}x^i h^{n-i}-x^n}{h} \\ &=\lim_{h\to 0}\frac{\sum_{i=0}^{n-1}{n\choose i}x^i h^{n-i}}{h} \\ &=\lim_{h\to 0}\sum_{i=0}^{n-1}{n\choose i}x^i h^{n-1-i} \\ &=\frac{n!}{(n-1)!1!}x^{n-1} \\ &=nx^{n-1}. \end{align*}$$