## ABSTRACT

This chapter explores how the Laplace transform interacts with the basic operators of calculus: differentiation and integration. The chapter begins with the derivation of what may be the most important identity of the chapter, the “transform of the derivative” identity. It is then shown how this identity can be used to convert a differential equation with initial values to an easily solved algebraic equation in which the solution is the Laplace transform of the solution to the original initial-value problem. This is followed by a “naïve” derivation of the “derivative of the transform” identity and a discussion of how this identity can be used to find the Laplace transforms of additional functions. All this naturally leads to the analogous development and use of two analogous identities — the “transform of an integral” and the “integral of a transform” identities. The chapter ends with an appendix on the rigorous verification the “derivative of a transform” identity.