What is the buffer equation and how is it derived?

Let's say we have an acid (HA) and its conjugate base (A^-) in a solution. The chemical formula is written as the following:

                                                      HA + H₂O ↔ H₃O⁺ + A^-

The buffer equation (Henderson-Hasselbalch equation) is derived for the above acid-conjugate base pair as shown by the following equations:

The last equation is the buffer equation (Henderson-Hasselbalch equation) that is used for buffer calculation of an acid - conjugate base pair.

If we have a base (B) and its conjugate acid (BH⁺) then the chemical formula is:

                                                       B + H₂O ↔ BH⁺ + OH^-

and the derivation is the following:

The last equation in the above is the buffer equation for a base-conjugate acid pair. Note that in the above derivation we used the relation pKw = pH + pOH.