I am in need of some assistance to solve problem 3.8 in Kirillov's An Introduction to Lie Groups and Lie Algebras. Copied below:
Let $SL(2, \mathbb{C})$ act on $\mathbb{C} P^1$ in the usual way: $$\begin{pmatrix} a & b \\ c & d \end{pmatrix} [x:y] = [ax+by:cx+dy].$$ This defines an action of $\mathfrak{g} = \mathfrak{sl}(2,\mathbb{C})$ by vector fields on $\mathbb{C} P^1$. Write explicitly the vector fields corresponding to $h, e, f$ in terms of the coordinate $t = x/y$ on the open cell $\mathbb{C} \subset \mathbb{C} P^1$.
I am aware that the action of $SL(2,\mathbb{C})$ on $\mathbb{C}P^1$ is equivalent to a map $\rho: SL(2,\mathbb{C}) \to \mathbb{C}P^1$, and further the pushforward $\rho_*$ defines the action of the Lie algebra. In terms of one-parameter subgroups, recalling $$ h = \begin{pmatrix}1&0\\ 0&-1\end{pmatrix},$$ $\exp(sh)$ is a curve in the Lie group with tangent vector $h$ at $s=0$, and so the action of $h$ on $\mathbb{C}P^1$ is defined to be $$ \frac{d}{ds}\exp(sh)([x:y])|_{s=0} = \frac{d}{ds} [e^s x: e^{-s} y]|_{s=0}$$ at which point I'm a little confused.
Edit: I should also add that this is a homework assignment, so hints are strongly preferred over full solutions.
You already have most of the solution. Now you just need to write the expression you got in means of the desired coordinate.
The complex coordinate on the set $U_y:=\{[x:y]|y\neq0\}$ is $$ \varphi:U_y\to\mathbb{C},\quad[x:y]\mapsto\frac{x}{y}. $$ This means that the path whose velocity you are after is given by $$ \gamma:(-\epsilon,\epsilon)\to\mathbb{C},\quad s\mapsto[e^sx:e^{-s}y]\mapsto e^{2s}\frac{x}{y}. $$
Can you take it from here?