I apologise for the confusion of the following sentences. I'm lazy to give more information about *Rough path theory* as Is a fairly broad subject.


On page 14 of *"A Course on Rough Paths
 With an Introduction to Regularity Structures*" by **Peter K. Friz & Martin Hairer** has written:


For $\alpha \in (1/ 3; 1 /2]$, define the space of $\alpha$-Hölder rough paths (over V ),
in symbols $\mathcal C^{\alpha} ([0,T]; V )$, as those pairs $(X; \mathbb X) =: \mathbf{X}$ such that
$$
||X||\_{\alpha}:= \sup\_{
s\neq t \in [0;T ]}
\frac{|X\_{s,t}|}{|t-s|^{\alpha}}
 < \infty , \quad ||X||\_{2\alpha}:=\sup\_{
s\neq t \in [0;T ]}
\frac{|\mathbb X\_{s,t}|}{|t-s|^{\alpha}}
 < \infty ,
$$
and such that the algebraic Chen relation ( is satisfed.


And on page 56 it hase written:
 Given a path $X \in \mathcal C^{\alpha}([0, T ]; V )$, we say that $Y \in \mathcal C^{\alpha}([0, T ]; \hat{W} )$, is controlled by $X$ if there exists $Y' \in \mathcal C^{\alpha}([0, T ]; \mathcal L(V , \hat{W})$, so that the remainder term $R^Y$ given implicitly through the relation $$ Y\_{s,t }= Y\_{s0} X\_{s,t} + R\_{s,t}^Y , $$ satisfes $||R^Y||\_{ 2 \alpha}< 1$. This defines the space of controlled rough paths, $(Y, Y') \in \mathcal D\_X^{2α}([0, T ]; \hat{W })$: Although $Y'$ is not, in general, uniquely determined from Y. We call any such $Y'$ the *Gubinelli derivative* of $Y$ (with respect to $X$). Here, $R\_{s,t}^Y$ takes values in $\hat{W}$, and the norm $|| \cdot||$.


**Question :** What is the intuition behind this idea of the Gubinelli derivative? 


Any help is appreciated with thanks in advance.



Intuition behind Gubinelli derivative
In a way it is very much like a usual derivative. Recall first that for a regular function $Y$, its derivative $Y'\_s$ at a point $s$ is the (unique) number such that
$$
Y\_{t,s}=Y'\_s(t-s)+ R\_{s,t},
$$
where $R\_{s,t}\to0$ faster than linearly. If $Y$ is twice differentiable, then $R\_{s,t}\lesssim |t-s|^2$. That is, as a function of $t$, $Y\_t$ "looks like" the linear function $Y\_s+Y'\_s(t-s)$, in the neighborhood of $s$.


Now simply replace the linear function by $X$. So we impose
$$
Y\_{t,s}=Y'\_sX\_{t,s}+R\_{s,t}
$$
with the remainder $R\_{s,t}\to0$ faster than the first term, that is, faster than $|t-s|^\alpha$ (The condition $R\_{s,t}\lesssim|t-s|^{2\alpha}$ from Friz-Hairer corresponds to the twice differentiable scenario in the previous case). Then as a function of $t$, $Y\_t$ "looks like" the path $Y\_s+Y'\_sX\_{s,t}$. This is great news for integration: we can of course integrate $Y\_s$ against $dX\_t$ (since as a function of $t$ it is just constant), and we can also integrate $X\_{s,t}$ against $dX\_t$ (by the definition of a rough path).


Actually, I wouldn't focus so much on assigning a meaning to $Y'$ itself, but rather focus on what the *existence* of a $Y'$ means for $Y$.


