Group delay of LTI system

Homework Statement

Show that the group delay of an LTI discrete-time system characterized by a frequency response $H(e^{j\omega})$ can be expressed as

$\tau(\omega)= Re\left\{\frac{j\frac{dH(e^{j\omega}}{d\omega}}{H(e^{j\omega}}\right\}$.

The Attempt at a Solution

I think I understand the rest of the proof (not shown here) but I don't see why the equation in (2) is valid. First, I had the idea that the first term in the equation in (2) is the real part and the second term in the equation is the imaginary part of the derivative, but it does not really appear to me to be like that.

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