We need to be somewhat careful about whether or not f' is well defined. We have to be certain that the limit exists before we claim that it is equal to anything. The existence of some limits is trivial, while others are a little more questionable. Suppose, for example,
$$f''=\frac{1}{\sqrt{x}}$$
First, consider the polynomial
$$P(x)=\prod_{k=0}^K(x+z_k)$$
Define the nth elementry symmetric function $e_n(Z)$ for the set of roots $Z=\{z_k\}_{k=0}^K$ to be
$$e_n(Z)=\sum_{1\leq i_1<i_2<\cdots<i_n\leq K} z_{i_1}z_{i_2}\cdots z_{i_n}$$
(Think of this as a sum of all possible -product-...