I would start with the definition of the derivative and see what you get:
$\displaystyle g'(x) = \lim_{h \to 0} \dfrac{g(x+h) - g(x)}{h}$
Let f(x) be a function of bounded variation on closed, bounded interval [a, b].
Let g(x) = V(f, [a, x])
where given some partition of [a, x].
How can I calculate the derivate of g(x)?
I know of differentiation term-by-term but not sure if this would apply here.
Thanks
No, quite the contrary. Instead, you just showed that if $g'(x)$ exists, then $g'(x) \ge |f'(x)|$ since you want the supremum over all possible partitions (one of which is the partition with just two points).