How to compute Gubinelli Derivatives

Let $$X,Y \in C^{\alpha}([0,1],\mathbb{R})$$ be $$\alpha$$-Hölder paths in $$\mathbb{R}$$ with $$1/3 < \alpha < 1/2$$. Then a path $$Y' \in C^{\alpha}([0,1],\mathbb{R})$$ is called a Gubinelli derivative of $$Y$$ with respect to $$X$$ if there exists a $$C > 0$$ s.t.

$$\vert Y_t - Y_s - Y_s' (X_t - X_s) \vert \leq C \vert t - s \vert^{2 \alpha}, ~~~ \forall s,t \in [0,1].$$

I am fine with the definition of Gubinelli derivatives, but I am struggling to compute it even for very simple functions. For example, let $$X_t := t^{\alpha}$$ and $$Y_t := t^{\beta}$$ with $$1/3 \leq \alpha \leq \beta \leq 1/2$$. What would be the Gubinelli derivative in this example?

Of course if $$\alpha = \beta$$, then $$Y_s = 1$$ for every $$s \in [0,1]$$ is sufficient.

I also know that once a Gubinelli derivative $$Y'$$ is found, then for a sufficiently regular function (say $$\phi \in C^2_b$$) one can find $$(\phi(Y))'$$ explicitely.

As mentioned in the section 6.2 in "A course in rough paths"-CRP for short, if you can find $$t_{n}\to s$$ such that

$$\frac{|X_{s,t_{n}}|}{|t_{n}-s|^{2\alpha}}\to +\infty$$

then the Gubinelli-derivative $$Y'$$ is uniquely determined (Proposition 6.4 in CRP). They have a formula for it:

$$Y'_{s}=\lim_{t_{n}\to s}\frac{Y_{t_{n}}-Y_{s}}{X_{t_{n}}-X_{s}}\label{1}\tag{1}$$

for a particular sequence $$t_{n}\to s$$.

So working in reverse, if you can show that the above expression $$Y'_{s}\in C^{\alpha}$$, then you got your Gubinelli-derivative and thus you got a rough-integral formulation (see Theorem 4.10 in CRP).

In the particular case of $$X_{t}=t^{\alpha}$$, the rough path lift is (Remark 2.2. in CRP)

$$\mathbb{X}_{s,t}:=\frac{1}{2}(X_{t}-X_{s})^{2}=\frac{1}{2}(t^{\alpha}-s^{\alpha})^{2}.$$

It is not truly rough because the limit goes to zero:

$$\frac{|X_{s,t_{n}}|}{|t_{n}-s|^{2\alpha}}\to 0.$$

So uniqueness seems unlikely. Now, using the remainder we see for $$t\approx s$$ and $$s$$ away from zero:

$$\frac{\vert Y_t - Y_s - Y_s' (X_t - X_s)\vert}{ \vert t - s \vert^{2 \alpha}}\approx \frac{ s^{\beta-1} \vert t-s\vert -Y'_{s} s^{\alpha-1} \vert t-s\vert}{ \vert t - s \vert^{2 \alpha}}\to 0$$

since $$\alpha<\frac{1}{2}$$. So any function $$Y'_{s}\in C^{\alpha}$$ will do eg. $$Y'_{s}=s^{\alpha}$$ for s away from zero.

However for $$s$$ close to zero or equal, we get $$t^{\beta-2\alpha}-Y'_{0}t^{-\alpha}$$ which diverges as $$t\to 0$$ even if we set $$Y'=0$$ because $$2\alpha>\frac{2}{3}>\frac{1}{2}>\beta$$.

If you want me to add more details, please let me know.