Define $\theta: R[x] \to R[x]$as $\theta(f)=f'$.
I understand the basics but I'm not sure I'm doing this correctly. I know first we have to show that the function is well defined, so we take 2 distinct elements from the domain. So would $x_1^2$, $x_2^2$ work? Where $(x_1)^2 = (x_2)^2$.
Then $((x_1)^2)'= ((x_2)^2)'$. So $\theta(x_1)=\theta(x_2)$.
I just feel like this is wrong. Any suggestions?