  The worldline method is a powerful numerical path-integral framework for
computing Casimir and Casimir-Polder energies. An important challenge arises
when one desires derivatives of path-integral quantities--standard
finite-difference techniques, for example, yield results of poor accuracy. In
this work we present methods for computing derivatives of worldline-type path
integrals of scalar fields to calculate forces, energy curvatures, and torques.
In Casimir-Polder-type path integrals, which require derivatives with respect
to the source point of the paths, the derivatives can be computed by a simple
reweighting of the path integral. However, a partial-averaging technique is
necessary to render the differentiated path integral computationally efficient.
We also discuss the computation of Casimir forces, curvatures, and torques
between macroscopic bodies. Here a different method is used, involving summing
over the derivatives of all the intersections with a body; again, a different
partial-averaging method makes the path integral efficient. To demonstrate the
efficiency of the techniques, we give the results of numerical implementations
of these worldline methods in atomplane and plane-plane geometries. Being quite
general, the methods here should apply to path integrals outside the worldline
context (e.g., financial mathematics).
