Show that $f(F(x))F'(x)$ is measurable.

This is a equation from Stein-Sharkarchi Real Analysis.

Let $F$ be absolutely continuous and increasing on $[a,b]$ with $F(a)=A$ and $F(b)=B$. Suppose $f$ is any measurable function on $[A,B]$.

Show that $f(F(x))F'(x)$ is measurable on $[a,b]$.

I am really having a hard time starting this problem. I know that $F'(x)$ is definitely measurable, but $f(F(x))$ need not be.