Let $X,Y$ be normed spaces, $L(X,Y)$ be the set of all bounded linear operators, $\Phi$ be a set of continuous functions from $\mathbb R$ to $X$.
Consider the following definition.
A function $f:X \rightarrow Y$ is said to be $\Phi$-differentiable if there is $D_\bullet:X \rightarrow L(X,Y)$ such that for all $\varphi \in \Phi$, $f(\varphi(t))=f(\varphi(0))+D_{\varphi(0)}(\varphi(t)-\varphi(0))+o(\| \varphi(t)-\varphi(0) \|)$, $t \to 0$.
It seems that $f$ is Fréchet (resp. Hadamard, Gâteaux) differentiable if and only if it is $\Phi$-differentiable with $\Phi$ being the set of all continuous (resp. differentiable, affine) functions. Differentiability at a point $x \in X$ can be defined restricting $\Phi$ to functions $\varphi$ satisfying $\varphi(0)=x$.
Does that make sense?