Derivation of formula for Hydrolysis constant
Sorry, I didn't get that. first of all, which hydrolysis are you talking about? I will talk about weak acid and strong base equilibrium. You have three concentrations [HA], [H+], [A-]. For instance, if they took 1L of 1 mM of HOAc (acetic acid) and added 0.3 mmoles of NaOH (concentrated, so no dilution) you get [A-]+[HA] = 1mM Then you say HA &lt;-> H+ + A- has Ka. Then

[H+][A-]/[HA] = Ka. Then you say  [HA] = x, so [A-] = concentration_total- x (in this case 1mM -x). Then you get an equation 
[H+]*[concentration_total-x]/x=Ka. Then you solve this square equation.

Regarding your question. I think you are mixing up to things: protonation of weak acid (stuff, relevant to analytical chemistry with a lot of drama, pH swings etc) and the behavior of other salts (they just dissociate completely and we forget about them). In stories like "to a solution of NaCl and K-acetat a drop of acid was added. What happen?" you should first forget about NaCl. It has no impact on the question. You should also forget about K+. Now you have "to a solution of Acetate- a drop of acid was added. What happen to pH?" Now you can easily calculate it using a square equation.

Run away from chemistry. Carcinogens, stinky chemicals, no  jobs... It teaches you how to make poisons and drugs, but how useful is that?