Given a formal parameter $\delta$ recall that the Temperley-Lieb algebra $\mathrm{TL}_n(\delta)$ is the unital $\Bbb{C}(\delta)$-algebra generated by symbols $U_1, \dots, U_{n-1}$ subject to the (defining) relations:
\begin{equation} \begin{array}{l} U_i^2 = \delta U_i \\ U_i U_{i \pm 1} U_i = U_i \\ U_i U_j = U_j U_i \quad \text{for} \quad \big|\, i - j \, \big| > 1 \end{array} \end{equation}
Question: View $\mathrm{TL}_n(\delta)$ as an infinite-dimension $\Bbb{C}$-algebra. Does there exist a $\Bbb{C}$-algebra derivation $D: \mathrm{TL}_n(\delta) \longrightarrow \mathrm{TL}_n(\delta)$ such that $D$ and ${d \over {d\delta}}$ coincide upon restriction to the central $\Bbb{C}$-subalgebra $\Bbb{C}(\delta) \subset \mathrm{TL}_n(\delta)$ ?
Remark: The answer seems to be YES for $\mathrm{TL}_{\,3}(\delta)$. An example of such a derivation is realised by setting
\begin{equation} \begin{array}{ll} D U_1 &\displaystyle = \ {1 \over {1 - \delta^2}} \ U_1 U_2 \, + \, {\delta \over {\delta^2 -1}} \ U_1 \\ D U_2 & \displaystyle = \ {1 \over {1 - \delta^2}} \ U_2 U_1 \, + \, {\delta \over {\delta^2 -1}} \ U_2 \\ \displaystyle D f &\displaystyle = \ { df \over {d \delta}} \quad \text{for all $f \in \Bbb{C}(\delta)$ } \end{array} \end{equation}
and extending by the (non-commutivative) Leibnitz rule. For $\text{TL}_{\, 3}(\delta)$ derivations of this kind are parametrised by a choice of $\tau \in \Bbb{C}(\delta)$; the choice in the example is $\tau = {\delta \over {\delta^2 - 1}}$.
best, Ines.
Too long for a comment, but I did a calculation and there is actually a 3 parameter family of derivations for $TL_3(\delta)$ (unless I made a mistake).
$$\begin{align} D U_1 &= \left( \frac{\delta}{\delta^2-1}+\alpha\right)U_1-\beta\left(\frac{1}{\delta^2-1}+\alpha\delta\right)U_1U_2-(1-\beta)\left(\frac{1}{\delta^2-1}+\alpha\delta\right)U_2U_1 \\ D U_2 &= \left( \frac{\delta}{\delta^2-1}-\alpha\right)U_2-\gamma\left(\frac{1}{\delta^2-1}-\alpha\delta\right)U_1U_2-(1-\gamma)\left(\frac{1}{\delta^2-1}-\alpha\delta\right)U_2U_1 \end{align}$$
Your example corresponds to $\alpha=0, \beta=1, \gamma=0$, but one can take any $\alpha,\beta,\gamma\in \mathbb C(\delta)$.
I have thoughts on how one might organize the computation for higher $n$, but unfortunately I am not familiar enough with these algebras to know if these computations can be fully carried out.