  We develop the representation theory for reductive linear differential
algebraic groups (LDAGs). In particular, we exhibit an explicit sharp upper
bound for orders of derivatives in differential representations of reductive
LDAGs, extending existing results, which were obtained for SL(2) in the case of
just one derivation. As an application of the above bound, we develop an
algorithm that tests whether the parameterized differential Galois group of a
system of linear differential equations is reductive and, if it is, calculates
it.
