  We consider a family of Markov chains whose transition dynamics are affected
by model parameters. Understanding the parametric dependence of (complex)
performance measures of such Markov chains is often of significant interest.
The derivatives of the performance measures w.r.t. the parameters play
important roles, for example, in numerical optimization of the performance
measures, and quantification of the uncertainties in the performance measures
when there are uncertainties in the parameters from the statistical estimation
procedures. In this paper, we establish conditions that guarantee the
differentiability of various types of intractable performance measures---such
as the stationary and random horizon discounted performance measures---of
general state space Markov chains and provide probabilistic representations for
the derivatives.
