I'd like to try to redefine the derivative (in a way equivalent to the usual definition) of a function $f: U \subseteq R \to R$ to make it clear that the derivative $Df(a)$ is the linear part of the best affine approximation to $f$ evaluated at $a$.
So I would think my proposed definition should be something like: For $a\in U$ yada yada yada $$f(a+h) = f(a) + Df(a)h + r(h)$$ where differentiability holds when $\lim_{h\to 0} \frac{r(h)}{h}=0$.
However I notice that this equation which I'd like to use to define the derivative $Df(a)$ is really already being used to define something: $r(h)$.
Is there some way I can adjust this or add more conditions or something so that I can define the derivative in a way that makes it apparent what the derivative is (as a part of an affine approximation)?