The motivation of this question is when I am looking at Taylor's Remainder theorem, suppose $f$ is $n+1$ differentiable, then the remainder can be written as $$R_{n+1}(x) = \frac{f^{(n+1)}(c)}{n!} x^{n+1} \quad \text{ where } c\in [0,x],$$ but I don't see $R_{n+1}(x) \in O(x^{n+1})$ if we have no control of $f^{(n+1)}(c)$ for $c\in [0,x]$.
I know since $f'$ is a pointwise limit of continuous functions, thus it is continuous on a dense set, would this help?
A counterexample to the question in the title is yielded by the function defined by $f(x)=x^2\sin(1/x^2)$ if $x\ne 0$ and $f(0)=0$. This is differentiable everywhere with $f'(0)=0$ (use the definition of the derivative). But if $m\in\mathbb{N}$, then $f'(m^{-1})=-2m\cos(m^2)+2\sin(m^2)/m$ and as $m\to\infty$, $\sin(m^2)/m\to 0$ and $\left|2m\cos(m^2)\right|$ oscillates wildly between $0$ and $2m$. So on any nbhd of $0$, the derivative is not bounded.