In some regression models, our interest is on estimating only some parameters instead of all the parameters in the model. Ds-optimality is used for this purpose. We start with a broad review of some important optimal design theories, some popular design criteria along with their properties. We then determine the optimality conditions for our optimization problems. After explicitly introducing the Ds-optimality and exploring its properties, we construct Ds-optimal designs for polynomial regression models in one and two design variables. We construct the optimal designs using a class of multiplicative algorithms, indexed by a function which depends on the derivatives of the criterion function. We also develop strategies for constructing Ds-optimal designs and investigate techniques for improving convergence rates by using the properties of the directional derivatives. Finally we provide some concluding remarks and a discussion of some potential future work.