Questions about the differential of Ad.

Question

In the book linear algebraic groups by Humphreys. Page 73, Lemma B says that for $x \in GL(n, K)$, $\mathbf{x} \in gl(n, K)$ we have $Ad x(\mathbf{x}) = x\mathbf{x}x^{-1}$. In the proof, the first line of the equations, it is said that for $T_{ij}$ in $K[GL(n, K)]$, we have $$Ad x(\mathbf{x})(T_{ij}) = \rho_x(*\mathbf{x})\rho_x^{-1}(T_{ij}) \quad (1)$$. But on Page 66, it is proved that $$Ad x(\delta) = \rho_x\delta \rho_x^{-1} \quad (2)$$. The formula (2) is different from (1) and (2) has no *. I am confused. Could you explain this? Thank you very much.