Suppose $f$ is a bounded variation function on $[a,b]$. $v(x) = TV([a,x])$ is the total variation of $f$ on $[a,x]$. From bounded variation of $f$, monotonicity of $v$, we know that $f$ and $v$ have derivative almost everywhere on $[a,b]$.
I'm now going to prove $|f'(x)| = v'(x) \quad a.e.$ I'm sure this is correct. But I can't find any reference in some most popular analysis textbooks. They only mention that $|f'(x)| \leq v'(x)$. How can I prove the other side of the inequality?