The techniques for computing $S$-matrix elements for nonpolynomial scalar-field Lagrangians with derivative interactions are presented. To second order in the interaction Lagrangian, it is shown that all the dependence arising from the derivative part is completely separated out as operators acting on integrals identical to those obtained in a nonderivative theory. The Fourier transforms of self-energy graphs for a class of non-local interaction Lagrangians are taken in the massless case. The on-mass-shell contributions are determined by the analytic continuation of the coefficients appearing in the series expansion of the Lagrangian. As special examples, two Lagrangians which are isoscalar analogs of chiral $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ Lagrangians are treated. The possible equivalence of on-mass-shell matrix elements for Lagrangians related by nonlinear field transformations is discussed.

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