To maintain the effectiveness of a buffer, relative concentrations of the acid and the conjugate base should not differ by a factor of 10. Based on this information, derive pH range within which the buffer can work effectively.
I know that I have to use the HH equation, but I cannot derive it.
You don't need to derive the HH equation, only derive the pH range needed.
pH = pKa + log(base)/(acid)
The problem tells you that acid and conjugate base should not differ by more than a factor of 10. Therefore, use
acid = 10*base as the lower end and 10*acid = base as the top end.
pH = pKa + log(base)/(10*base)
pH = pKa + log (1/10)
pH = pKa -1 OR for the other one
pH = pKa + log(10*acid)/(acid)
pH = pKa + log(10/1)
pH = pKa + 1
You can put these two togther so the equation becomes
pH = pKa +/- 1.