Consider the action of the cyclic group $C_p$ on the projective line over a field of characteristic $p$ via $a\cdot[x:y]=[x+ay:y]$. I am asked to describe the tangent space at infinity, and prove that the action of $C_p$ on the tangent space is trivial.
Now I know that the tangent space is one-dimensional since the projective line is a nonsingular variety, and hence the tangent space is isomorphic to the ground field.
I am not sure how to handle the other part of the question. A hint suggests that $C_p$ acts linearly on the tangent space, hence the action is just multiplication by a $p$th root of unity, i.e. $1$ in this case. I do not understand how $C_p$ is expected to act linearly on the tangent space.
If you have a map of varieties $f:X\to Y$ sending $x\mapsto y$, you get a linear map on tangent spaces $df_x:T_xX\to T_yY$ called the differential. In particular, if you have a group action on $X$ by $G$ and $x$ is a fixed point, then you get an action of $G$ on $T_xX$ by $g$ acts as $dg_x$. Since $d(-)$ is a functor, we have that it respects composition and identites, and in our case we see that $d(a\cdot)^p=Id_k$. From here, the hint finishes the problem: $d(a\cdot)$ acts as a $p^{th}$ root of unity on $k$, which is necessarily just $1$.