Numerical differentiation is undoubtedly a fundamental problem in signal processing and control engineering, due to its countless applications. The goal of this paper is to address this question within an algebraic framework. More precisely, we consider a noisy signal and its orthogonal polynomial series expansion. Through the algebraic identification of the series coefficients, we then propose algebraic differentiators for the signal. Examples based on Hermite and Laguerre polynomials illustrate these algebraic differentiators.