= Solution
With sufficiently many <derivatives>, <Taylor's theorem> gives a finite approximation and a controlled remainder. For example, if $f$ is $N+1$ times continuously differentiable between $a$ and $a+\delta$, then
$$
\boxed{f(a+\delta)=\sum_{j=0}^N\frac{f^{(j)}(a)}{j!}\delta^j
+\frac{f^{(N+1)}(a+\theta\delta)}{(N+1)!}\delta^{N+1}},\qquad0<\theta<1.
$$
Thus <derivatives> at the nearby base point determine the <polynomial> approximation; a bound on the next <derivative> bounds its error. The smallness of $\delta$ makes this useful, provided the <derivatives> remain controlled.
The statement's bare assumption of differentiability alone guarantees only $f(a+\delta)=f(a)+f'(a)\delta+o(\delta)$. Higher-order formulas need higher regularity. Even smoothness does not by itself guarantee equality with an infinite <Taylor series>; that stronger conclusion requires the remainder to tend to zero.
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