Actually, one differentiates the polynomial group representation $\sigma_d : \operatorname{GL}_2(\mathbb{Q}) \to \operatorname{GL}(E^{\otimes d})$ at the identity of $\operatorname{GL}_2(\mathbb{Q})$ to obtain a representation $\rho_d : \mathfrak{gl}_2(\mathbb{Q}) \to \mathfrak{gl}(E^{\otimes d})$ of the associated Lie algebra. For $g \in \operatorname{GL}_2(\mathbb{Q})$, $x \in \mathfrak{gl}_2(\mathbb{Q})$, and $u_1, \ldots, u_d \in E$, we have
$$ \sigma_d(g)(u_1\otimes \cdots \otimes u_d) = g(u_1) \otimes \cdots \otimes g(u_d),$$
$$\rho_d(x)(u_1 \otimes \cdots \otimes u_d) = \sum_{k=1}^d u_1 \otimes \cdots \otimes u_{k-1} \otimes x(u_k) \otimes u_{k+1} \otimes \cdots \otimes u_d.$$
The extension of $\rho_d : \mathfrak{gl}_2(\mathbb{Q}) \to \mathfrak{gl}(E^{\otimes d})$ to a representation $\tilde\rho_d : U(\mathfrak{gl}_2(\mathbb{Q})) \to \operatorname{End}(E^{\otimes d})$, whose existence and uniqueness is guaranteed by the universal property of the inclusion $\mathfrak{gl}_2(\mathbb{Q}) \to U(\mathfrak{gl}_2(\mathbb{Q}))$, can also be described as follows. Let $(x_1,x_2,x_3,x_4)$ be an ordered basis of $\mathfrak{gl}_2(\mathbb{Q})$. Then, by the Poincaré-Birkhoff-Witt theorem, the (non-commutative) monomials of the form $x_1^{k_1}x_2^{k_2}x_3^{k_3}x_4^{k_4}$, where $k_j \in \mathbb{Z}_{\geq0}$, form a basis of $U(\mathfrak{gl}_2(\mathbb{Q}))$ over $\mathbb{Q}$. The representation $\tilde\rho_d$ is defined on basis elements via the rule
$$ \tilde\rho_d(x_1^{k_1} \cdots x_4^{k_4}) = \rho_d(x_1)^{k_1} \cdots \rho_d(x_4)^{k_4},$$
and then extended by linearity to the whole of $U(\mathfrak{gl}_2(\mathbb{Q}))$. The product on the right hand side in the above formula is taken in the associative algebra $\operatorname{End}(E^{\otimes d})$ having the same underlying vector space as the Lie algebra $\mathfrak{gl}(E^{\otimes d})$.
