What is the motivation of the $L^p$ differentiability? I was reading some papers and come up with the next definition :

A function is differentiable in the $L^p$ sense at $x$ if there
  exists a real number $f'_p(x)$ such that $$\bigg(\frac{1}{h}∫_{-h}^{h}|f(x+s)−f(x)−f_p'(x)s|^pds\bigg)^{1/p}=o(h)$$

And he states that many $f_p'$ are equivalent to an ordinary derivative on an almost everywhere basis. I saw this kind of differentiability is studied in some papers, but I don't know where does it started, i.e. what was it's motivation to define such operation and function space.
Ash, J. Marshall, An (L^p) differentiable non-differentiable function, Real Anal. Exch. 30(2004-2005), No. 2, 747-754 (2005). ZBL1107.26010.
 A: I don't know the literature, but it maybe helpful to understand exactly what this notion of differentiability gains for us. 
First, unlike the Sobolev notions, this does not handle functions which belong in Holder classes. For example, based on the definition the absolute value function is not $L^p$ differentiable for any $p$, but it is certainly (locally) in the Sobolev class $W^{1,p}$ for every $p\in [1,\infty]$. 
In fact, whenever $f$ is a function such that the one sided derivatives from the left and from the right independently exist, but do not agree, such a function is not $L^p$ differentiable in the sense defined in the paper. 
So what does $L^p$ differentiability gain for us? It gains when your function fails to be differentiable near $x$ due to it being oscillatory in a certain way. An example:
Let the set $A = \cup_{k = 10}^\infty  [ 1/k - 2^{-k}, 1/k + 2^{-k}]$, and let $f$ be the indicator function of $A$. This function is clearly not differentiable at $x = 0$. However, since 
$$ \int_{-h}^h |f(s)|^p ~ds \leq  \sum_{k = \lfloor 1/h \rfloor }^\infty 2^{1-k} = 2^{2-\lfloor 1/h \rfloor } $$
we see that for any $p\in [1,\infty)$, $f$ is $L^p$ differentiable at the origin with derivative $0$. 
As to why measuring things with respect to $L^p$ means can be useful: there is a big hint in the paper of Calderon and Zygmund, concerning elliptic PDEs. They wrote [emphases mine]:

It seems that the notion of differentiability which is most suited to the treatment of the problems that concern us, is not the classical one. It appears that it is more convenient to estimate the remainder of the Taylor series in the mean with various exponents. This type of differentiability is much more stable ...

If you are familiar with some harmonic analysis, what this is screaming out is that a lot of analytic estimates (Sobolev regularity for elliptic PDEs, singular integrals, etc.) work generally for functions measured on the $L^p$ scale, but often fail (just by a little) at $L^\infty$ (and sometimes $L^1$). 
You can of course ask why did Calderon and Zygmund not use the Sobolev class: the key is that they want to understand pointwise estimates of differentiability. As mentioned above, Sobolev classes are not so sensitive to pointwise properties. 
