How to compute the order $\text{ord}_P (f)$ for $f \in K(C)$

First lets fix some notation. Let $C$ be a projective curve (i.e. projective variety of dimension 1) defined over a field $K$. Suppose that $P \in C$ and that $P$ is a smooth point. It is known that in this case the local ring of $C$ at $P$, $K[C]_P$ is a discrete valuation ring with valuation given by

$$ \text{ord}_P (f) := \sup{\{d \in \mathbb{Z} \mid f \in M_{P}^{d} \}} $$

where $M_P$ is the maximal ideal of $K[C]_P$ and this definition can be extended to any function $\dfrac{f}{g} \in K(C)$ by

$$ \text{ord}_P \left ( \frac{f}{g} \right ) = \text{ord}_P (f) - \text{ord}_P (g) $$

Now the problem for me is that this definition literally does not let me compute anything. It just does not seem easy to me to compute the order of a function by using this definition and this is causing me trouble for understanding how to compute the divisor associated to a function and also how to compute differentials since I need to be able to compute the order of functions at the points in the curve.

On the other hand, for the particular case of the curve $\mathbb{P}^1$, since its function field $K(\mathbb{P}^1) \cong K(t)$ is so nice, I'm able to compute the order of a function relatively easy by looking at the order of the zero or pole of the corresponding rational function in one variable.

I would be very grateful if someone can explain how can one compute the order of a function at some point. The examples given in Silverman's book The Arithmetic of Elliptic Curves (which is the book I'm reading) just give the result but don't provide any explanation.

Example

This example is very important for me to understand since it comes again in the chapter about the Geometry of Elliptic Curves, so if someone can explain how to compute at least the orders calculated there, I guess that I can pick up from there.

Assume that $\text{char}(K) \neq 2$, and let $e_1, e_2, e_3 \in K$ be distinct. Consider the curve

$$ C: y^2 = (x - e_1)(x - e_2)(x - e_3) $$

Then if we denote the points $P_i := (e_i, 0)$ we have

$$ \text{div}(x - e_i) = 2(P_i) - 2(\infty) $$

and also

$$ \text{div}( y ) = (P_1) + (P_2) + (P_3) - 3(\infty) $$

I would really appreciate any help with understanding either this example or with how to compute the order of a function in general.

Thank you very much in advance.

Okay, once you know how to find local parameters everything else should be much easier. I encourage you to prove the following as an exercise.

Let $f(x, y) = 0$ be an affine curve and $p = (x_0, y_0)$ a point on it. If $\frac{\partial f}{\partial y}(x_0, y_0) \neq 0$, then $x - x_0$ is a local parameter at $p$; if $\frac{\partial f}{\partial x}(x_0, y_0) \neq 0$, then $y - y_0$ is a local parameter at $p$.

For intuition you should try to visualize how this works in the case $f(x, y) = x^2 + y^2 - 1$. Note that if $p$ is smooth then at least one of the above partial derivatives is nonzero.