Let $Aut(\mathbb{C} \mathbb{P}(1))$ be the Automorphism Group of $\mathbb{C} \mathbb{P}(1)$ ($\mathbb{C} \mathbb{P}(1)=$ the Complex Projective Space of dimension one).
Let $T \in Aut(\mathbb{C} \mathbb{P}(1))$, i.e., $T:\mathbb{C} \mathbb{P}(1) \longrightarrow \mathbb{C} \mathbb{P}(1)$ is a biholomorphism.
Let $p\in \mathbb{C} \mathbb{P}(1)$ such that $T(p)=p$ (i.e., $p$ is a fixed point for $T$).
What is the multiplier of the biholomorphism $T:\mathbb{C} \mathbb{P}(1) \longrightarrow \mathbb{C} \mathbb{P}(1)$ at the fixed point $p\in \mathbb{C} \mathbb{P}(1)$?
The following is taken from “§3. Fatou and Julia: Dynamics on the Riemann Sphere.” in Milnor: Dynamics in one complex variable: introductory lectures.
Let $S$ be a Riemann surface, $f:S \to S$ a non-constant holomorphic mapping, and $f^{\circ n}: S \to S$ its n-fold iterate.
$n$ distinct points $$ z_0, z_1, \ldots, z_{n-1}, z_n = z_0 $$ are called a “periodic orbit” or “cycle” if $f(z_j) = z_{j+1}$ for $0 \le j < n$, the number $n$ is the “period.” If $S$ is the complex plane or a subset thereof then $$ \lambda = (f^{\circ n})'(z_j) = f'(z_1)f'(z_2) \cdots f'(z_n) $$ is the “multiplier” or “eigenvalue” of the orbit.
In the more general case of self-maps of an arbitrary Riemann surface the multiplier is defined using a local coordinate chart around any point of the orbit.
In your case: If we represent the complex projective line as extended complex plane $\hat{\Bbb C} = \Bbb C \cup \{ \infty \}$ then its holomorphic self-maps are exactly the rational functions (and the automorphisms are the rational functions of degree one, that are the Möbius transformations).
If $p \in \Bbb C$ is a fixed-point of $f$ then $\lambda = f'(p)$ is the multiplier. If $p=\infty$ is a fixed-point of $f$ then its multiplier is the derivative of $g(z) = 1/f(1/z))$ at $z=0$.