How to show that $(\delta'f)(e)=\delta(\phi^* f)(e)$?

How to show that $(\delta'f)(e)=\delta(\phi^* f)(e)$?

I am reading linear algebraic groups.
I have a question in line 4 of the third paragraph of Page 66. How to show
that $(\delta'f)(e)=\delta(\phi^* f)(e)$ for all $f\in K[G]$? Here $G$ is
an algebraic group, $K$ is an algebraic closed field, $\delta' =
\operatorname{Ad} x(\delta) = d(\operatorname{Int} x(\delta))$,
$\phi=\operatorname{Int} x$, $\operatorname{Int} x(y) = xyx^{-1}$, $x, y
\in G$. Thank you very much.
I think that $\delta'f = d(\operatorname{Int} x(\delta))f$.