Using the chain rule,
Differentiating (a time shifted version of) the eqn.1.2 wrt
Finally we get the required probability, by substituting for in eqn.1.22 (keeping in mind that in this case), which is obtained by substituting eqns.1.28 and 1.24 in eqn.1.27.
Usually this is given the following form, by first substituting for from eqn.1.21 and then substituting from eqn.1.14.
If the continuous densities are used then can be
found by further propagating the derivative using the chain rule.
The same method can be used to propagate the derivative (if necessary) to a front end processor of the HMM. This will be discussed in detail later.