Yes, as long as a function of a complex variableisdifferentiable, then it can be differentiated using the usual differentiation rules.

"Differentiability" is a stricter consideration than for functions of a real variable, largely because the complex plane is two dimensional, but as long as you avoid things like separating real and imaginary parts and complex conjugate, you won't have any problem. In particular, polynomial functions and rational functions of the complex variable, z, are differentiable as long as the denominator is not 0 and all the usual differentiation rules apply.