This paper introduces the concept of derivatives of regular expressions and their properties, leading to the construction of state diagrams from regular expressions containing any number of logical operators. Regular expressions are defined using three operators: union, concatenation, and iterate, along with Boolean functions. The derivative of a regular expression with respect to a sequence is defined as the set of sequences that can follow the given sequence in the original expression. The paper presents algorithms for computing derivatives of regular expressions and discusses their properties, including that the derivative of a regular expression is itself a regular expression. The use of derivatives allows for the natural construction of state diagrams for sequential circuits. The paper also discusses the equivalence of states in automata and how derivatives can be used to determine this equivalence. The results are extended to multiple-output circuits, where each output is represented by a regular expression. The paper concludes that the derivative approach provides a powerful method for analyzing and constructing state diagrams for sequential circuits.This paper introduces the concept of derivatives of regular expressions and their properties, leading to the construction of state diagrams from regular expressions containing any number of logical operators. Regular expressions are defined using three operators: union, concatenation, and iterate, along with Boolean functions. The derivative of a regular expression with respect to a sequence is defined as the set of sequences that can follow the given sequence in the original expression. The paper presents algorithms for computing derivatives of regular expressions and discusses their properties, including that the derivative of a regular expression is itself a regular expression. The use of derivatives allows for the natural construction of state diagrams for sequential circuits. The paper also discusses the equivalence of states in automata and how derivatives can be used to determine this equivalence. The results are extended to multiple-output circuits, where each output is represented by a regular expression. The paper concludes that the derivative approach provides a powerful method for analyzing and constructing state diagrams for sequential circuits.