I've performed an ordinary least squares on a data set with one variable. For simplicity, let's say I've fitted a polynomial function $$f(x)=a+bx+cx^2+dx^3.$$ I obtain the best fit and the standard error (via the covariance matrix) for each of the fit parameters. Now, I want to calculate the error in $f'(x)=\frac{df(x)}{dx}$ at a particular value of $x$.
How should I go about doing this? I've taken a look at this post, but I don't quite understand how to apply it to $f'(x)$.
