If $f \in BV[a, b]$, show that $f'$ exists and is integrable.
My Attempt : I know that for any $f \in BV[a, b]$, we can write it as difference of two monotonic increasing functions and monotonic increasing functions are differentiable. But not sure if this is rigorous and correct.
Yes this is correct. Moreover since for a non decreasing function $g$ one has $\int_{[a:b]} g' d\lambda\leq g(b)-g(a)$ you obtain that $\int |f'| d\lambda \leq V(f)$ where $V(f)$ is the total variation of $f$. So its derivative is indeed integrable.
Note that the inequality can be strict : think of the cantor function.