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)?

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    $\begingroup$ What characterises the derived number at $a$ is $f(a+h)-f(a)-Df(a)h$ is the only affine function that is $o(h)$ in a neighbourhood of $0$. That is even the basis of the definition of the Fréchet derivative in topological vector spaces. $\endgroup$ – Bernard Nov 20 '15 at 2:20
  • $\begingroup$ Hmm.. I haven't learned about little o notation yet. It's on the list of things I still need to get to (behind big O notation incidently). $\endgroup$ – user291822 Nov 20 '15 at 2:23
  • $\begingroup$ It simply means what you've written about $r(h)/h$. $\endgroup$ – Bernard Nov 20 '15 at 2:24
  • $\begingroup$ Little-oh means that it is small, big-oh that it is bounded. Big-oh is usually more precise (e.g., $O(h^2)$ as opposed to $o(h)$), but little-oh is often good enough and sometimes all you can get. $\endgroup$ – marty cohen Nov 20 '15 at 4:10

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