Value of the derivative of a function at a point depends only on the germ at that point

Question

Suppose that f : I → R is a $C^∞$ function defined on an open subset I ⊆ R. How can I show that for $a \in I$ the value $f^n (s)$, n = 1, 2, 3, . . . of the derivative of $f$ of order n at s depends only on the germ of $f$ at s?

Answer 1

This will follow by induction if you can show that for any function $f$, its derivative evaluated at $s$ depends only on the germ of $f$ at $s$. This is clear since the derivative is $$\lim_{x\to s} \frac{f(x)-f(s)}{x-s}$$ which clearly only depends on the values of $f(x)$ in a small neighborhood of $s$.