Abstract: The investigation focuses on optimal design issues related to the comparison of regression curves within the framework of the Copula model. The μ_p-optimal is defined for comparing regression curves using a bivariate Copula model.An equivalence theorem is established using directional derivative.This theorem provides a theoretical foundation for solving and verifying μ_p-optimal designs. As an example, the study considers a scenario where marginal regression curves are modeled as low-order polynomial functions. The errors are assumed to follow a Gaussian distribution. The joint distribution is specified using a bivariate Gaussian Copula.Solutions for the μ₁-optimal design and the μ_∞-optimal design are derived. These results demonstrate the practical utility of the equivalence theorem in optimal design problems.