In classical mathematics, Newton-Leibniz differential operators determine coefficients in Taylor series. At the same time, there are relationships between Fourier coefficients of a (differentiable) function and its derivative. By the analogy, Boolean differential operators are viewed as coefficients of Taylor-Maclaurin series-like expressions for switching functions, usually dented as Reed-Muller expressions. Spectral interpretation of these expressions, permits to relate the Boolean difference to the coefficients in Fourier series-like expressions for switching functions.

This paper considers these two possible ways of introduction of differential operators for multiple-valued (MV) functions. We defined the Logic derivatives and Gibbs derivatives for MV functions as coefficients in Taylor-Maclaurin series for MV functions and through relationships to Fourier series-like coefficients, respectively.