Algebraic tools such as derivations play important roles in some central problems in affine algebraic geometry. The interest in images of differential operators and derivations arose from the research of the Jacobian conjecture in the last decade. In this talk, we focus on the problem of whether the images of certain differential operators and derivations are the so-called Mathieu-Zhao subspaces (a generalization of ideals). We will introduce the background, known results, and some of our work on this problem.