  We discuss an alternative approach to Fr\'echet derivatives on Banach spaces
inspired by a characterisation of derivatives due to Carath\'eodory. The
approach allows us to reduce many questions of differentiability to a question
of continuity. We demonstrate how that simplifies the theory of
differentiation, including the rules of differentiation and the Schwarz Lemma
on the symmetry of second order derivatives. We also provide a short proof of
the differentiable dependence of fixed points in the Banach fixed point
theorem.
