Can we say that, Since the given graph could be increasing and decreasing both. So $f'(x)$ could be positive and negative both so summation of $f'(x)$ over $x\in R$ could be infinite. Hence $f'(x)$ is unbounded.
1 - You can't really sum over a continuous set like $\mathbb{R}$. You integrate over it (integrals are limits of summations).
2 - How does the area under $f'$ being infinite relate to $f'$ being unbounded !?
The correct explanation is: $f'$ can be unbounded without being infinite/undefined at any point (thus making $f$ differentiable everywhere), if $f'$ tends to an infinite (positive or negative) as it approaches $-\infty$. That is, $\displaystyle \left | \lim_{x\to -\infty} f'(x) \right | = \infty$.
This way, $f'$ exists, and is finite on every value of $x \in \mathbb{R}$, but is still unbounded.