Could someone tell me where I might find information on deriving the standard error used in confidence and prediction intervals of y at a given value of x on a regression line. I can't find anything on the internet. Thanks for any help.
In the regression model, knowing that Y is Normally distributed, one can use properties of the multivariate normal distribution to provide an answer.
I would recommend you look at Wikipedia's comment, especially for example, in the case of the bivariate normal distribution, and in particular, the conditional distribution of Y given X.
There is also an associated expression for the variance of the conditional distribution, which answers your question. See also, summary presented on Page 4 of this educational reference.
Note, for a prediction interval, one adds an added sigma square term.