  Special functions like the polygamma, Hurwitz zeta, and Lerch zeta functions
have sporadically been connected with the nth derivatives of trigonometric
functions. We show the polylogarithm $\text{Li}_s(z)$, a function of complex
argument and order $z$ and $s$, encodes the nth derivatives of the cotangent,
tangent, cosecant and secant functions, and their hyperbolic equivalents, at
negative integer orders $s = -n$. We then show how at the same orders, the
polylogarithm represents the nth application of the operator $x \frac{d}{dx}$
on the inverse trigonometric and hyperbolic functions. Finally, we construct a
sum relating two polylogarithms of order $-n$ to a linear combination of
polylogarithms of orders $s = 0, -1, -2, ..., -n$.
