  Let $X$ be a perfect, compact subset of the complex plane, and let
$D^{(1)}(X)$ denote the (complex) algebra of continuously
complex-differentiable functions on $X$. Then $D^{(1)}(X)$ is a normed algebra
of functions but, in some cases, fails to be a Banach function algebra. Bland
and the second author investigated the completion of the algebra $D^{(1)}(X)$,
for certain sets $X$ and collections $\mathcal{F}$ of paths in $X$, by
considering $\mathcal{F}$-differentiable functions on $X$.
  In this paper, we investigate composition, the chain rule, and the quotient
rule for this notion of differentiability. We give an example where the chain
rule fails, and give a number of sufficient conditions for the chain rule to
hold. Where the chain rule holds, we observe that the Fa\'a di Bruno formula
for higher derivatives is valid, and this allows us to give some results on
homomorphisms between certain algebras of $\mathcal{F}$-differentiable
functions.
