The chain rule for F-differentiation

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 ([3]) investigated the
completion of the algebra D(1)(X), for certain sets X and collections F of paths in X, by considering 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 F-differentiable functions.