An alternative approach to Fr\'echet derivatives

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.