Measuring the smiling morn

I've finally got round to quantifying the changes in day length — something I often find myself pondering whilst on my early morning run, especially at this time of year when it seems to change so rapidly, but which ceases to interest me once I get home. After deciding to do something about it, I ran a couple of equations through Grapher to generate a curve showing day length:

Then it was a simple matter to determine the differential and convert the units from hours to minutes:

Now that I've worked out the numbers and confirmed my observations about the darkening mornings, I need never worry about any of this ever again. Hurrah!