Where to find derivation of s'' (stability factor for beta) for BJTs?
It seems I made some error in the manipulation of variables which is why I didn't get the same formula as in the book. I thought I needed to find partial derivatives, but it turns out that algebra is all that is needed. I'll just place the first few equations of the derivation if someone else wants to know since it's kind of long.

For an emitter-bias configuration:
$$V_{CC} - \frac{I_C}{\beta} R_B - V_{BE} -\frac{\beta + 1}{\beta} I_C R_E  = 0$$
$$I_C = \frac{\beta (V_{CC} - V_{BE})}{R_B+(\beta + 1)R_E} \tag1$$

Since \$s''= \frac{\Delta I_C}{\Delta \beta}\$, compute \$\Delta I_C = I_{C_2} - I_{C_1}\$:
$$\Delta I_C = I_{C_2} - I_{C_1} = \frac{\beta_2 (V_{CC} - V_{BE})}{R_B+(\beta_2 + 1)R_E} -\frac{\beta_1 (V_{CC} - V_{BE})}{R_B+(\beta_1 + 1)R_E}$$

Simplifying (a lot) and then using eq. 1 and then dividing \$\Delta I_C\$ by \$\Delta \beta\$ you will get the formula:
$$s'' = \frac{I_{C_1}(R_B + R_E)}{\beta_1(R_B + (\beta_2+1)R_E)}$$