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.