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I recently asked this Do all the curves become straight line in limit which led me to this question.

When we talk about Differentiation, in fact we have already assumed that if we zoom too much, then our curve tends to a straight line.

So, my question is, why !? Why should we accept this assumption? What if it really doesn't?

You may say it's totally Obvious but, somebody can say it's not. if we don't accept that every curvature tends to a straight line in limit, then speaking of Differentiation will be Nonsense, so, Is Differentiation build on unproved Assumption?

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    Differentation is not defined geometrically, considering curvature of curves. – 2017-01-21
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    Differentiation of a function $f$ is "built on" the requirement that $f$ is dfferentiable. There exist lots of functions that are not differentiable; they do not have derivatives, and zooming in on them will not tend toward straight lines. To say that $f$ is differentiable at a point $a$ is exactly to say that there is a linear map as in E.Joseph's answer, and that is the precise formulation of "zooming in tends to a straight line." – 2017-01-21
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    See e.g. Ethan Bloch, [The real numbers and real analysis](https://books.google.it/books?id=r0qcU9U2_I4C&pg=PA183) (2011), page 183 : "**Definition 4.2.1**. Let $I \subseteq \mathbb R$be an open interval, let $c \in I$ and let $f : I \to \mathbb R$ be a function. The function $f$ is **differentiable** at $c$ if ..." – 2017-01-21

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Differentiation is usually not defined geometrically but algebraically, using linear maps.

The differential of $f$ at the point $a$ is the linear map $\mathrm d_a(f)$ such that

$$f(a+h)=f(a)+h\mathrm d_a(f)+o(\vert \vert h\vert \vert).$$

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    Sorry but I don't catch what you mean – 2017-01-21