CI or SE of x-value occurring at Ymax of non-linear regression

Hello. I need some help to determine standards error or confidence interval associated with a x-value calculated at the Ymax of a 2nd order polynomial regression.

My data set is growth and decay over time. It is actually measuring the brightness of a light as it grows and then dims. I have successfully fit a 2nd order polynomial to this data with high r-squared. I am using Excel to do this.

I am ultimately trying to test the time to peak brightness as measured in different wavelengths. Each wavelength is a different data set. And each wavelength has its own 2nd order polynomial regression, with its own x-value corresponding to its Ymax. Let's call this value: Xmaxbright

To find Xmaxbright I took a derivative of my polynomial regression, and determined my x-value at Ymax. So far, I am in good shape. I have the Xmaxbright for the Ymax measured in red wavelength. And I have Xmaxbright for the Ymax measured in blue wavelength.

How can I determine the the standard error or confidence interval associated with each Xmaxbright? I need this in order to really know if there is a statistical difference between the Xmaxbright value for the red vs. blue light.

Excel gives me the error of the coefficients (along with a number of other stats) for my polynomial regression, but I am unsure how to use these, in order to determine some measure of confidence or error associated with the calculated Xmaxbright for red light vs. Xmaxbright for blue light.

Thanks so much for your help!


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Search for the "delta method". Here is Dasons writing, including example and code. Enjoy!
The great thing about that particular blog post is that it's incredibly on point for what @broudyt wants to do and yet probably not useful unless they plan on using R. I don't think I ever listed the formula for the specific case. With that said I don't know if excel gives the full covariance matrix for the estimate parameters? But we do need the covariance between some of the estimated parameters so that's a required first step if they're going to do this 'by hand' using the delta method.