analog values to ppm
When you calculate the derivative, the resulting units are the units of y divided by the units of x. So here, it would be \$V/K\$ (deg C and K are the same unit when considered differentially). If you divide the derivative by the typical/nominal voltage, the resulting units are then just \$1/K\$. The values will be fairly small, so you often use a unitless scale factor like ppm. Unitless scale factors are not really part of the SI system of units (as far as I know), but they are a common convention.

You are dealing here with unitless scale factors. The most common example of a unitless scale factor is %, meaning \$10^{-2}\$.

Your second plot shows a scale factor of "milli", meaning \$10^{-3}\$. I'm not sure if you have already divided by the nominal voltage or not. This is a fairly unusual scale factor when not combined with a base unit, but it's still done at times. 

You can convert milli to ppm by multiplying by 1000, since ppm means parts per million, \$1/1000000\$ = \$10^{-6}\$, which is 1000 times smaller than \$1/1000\$.

So a value of "0.4 milli" is the same as "400 ppm". Just make sure you are actually dealing with "milli"/K, first, not millivolts/K.