Why is this derivation well-defined?

Question

I'm reading Fulton's algebraic curves (page 105) and I'm trying to prove $\tilde D$ is well-defined:

Let's define $\tilde D$ as $\tilde D(z)=y^{-1}(Dx-zDy)$, then if $z=\frac{x_1}{y_1}=\frac{x_2}{y_2}$, we have

$$\tilde D(z)=y_1^{-1}(Dx_1-zDy_1)\ \text{and}\ \tilde D(z)=y_2^{-1}(Dx_2-zDy_2)$$

How can we prove these expression are equal?

Thanks

Answer 1

We have $x_1y_2=x_2y_1$. Apply $D$ on both sides and notice that the result states precisely that your two expressions are equal.

By the way: You should include more context in your question. What kind of derivative is $D$? From your other question (https://math.stackexchange.com/questions/1156918/bad-notation-in-fultons-algebraic-curves-book) I could get the information that $R$ is supposed to be a $k$-algebra and $D$ is a $k$-derivation.