high pass circuit as differentiator
I believe I've answered this elsewhere but, regardless, here is the way to look at this.  The transfer function for the RC HPF is:

\$\dfrac{j \omega RC}{1 + j \omega RC} \$

The "trick" is to look at the behaviour at low enough frequency such that \$ j \omega RC &lt;&lt; 1\$.  When this holds, the denominator is effectively just \$1\$ and the transfer function is effectively:

\$ j \omega RC\$

But, this is the transfer function for a differentiator with gain equal to \$RC\$.  That's really all there is to it.  For frequencies well below the corner frequency, the output is effectively proportional to frequency just as we would expect from a differentiator.

You're probably misunderstanding "stop band" in this context.  A 1st order high pass filter has a gentle roll-off that is just about 20dB / decade.  Signals aren't "stopped" below the corner frequency, they're increasingly attenuated.