You could plug in x+ϵω and take the standard part as the answer and each power of ϵ to be that order of derivate.
If one is already dealing in infinidesimals the fact of separate archimedian fields means the standard part does not mix with the infinidesimals as long as the functions of interest are tame “standard only” mechanics (ie real polynomials and such)
It could also be interesting to use the big side to maybe cancel some unwanted infinities. That is 2ω+3 is no fun as a end answer and ω+1 is no fun as answer but 2ω+3ω+1=2ωω+1+3ω+1=2−y+3ϵ−z where y and z are smaller than standard so it has a usable standard part answer of 2.
You could plug in x+ϵω and take the standard part as the answer and each power of ϵ to be that order of derivate.
If one is already dealing in infinidesimals the fact of separate archimedian fields means the standard part does not mix with the infinidesimals as long as the functions of interest are tame “standard only” mechanics (ie real polynomials and such)
It could also be interesting to use the big side to maybe cancel some unwanted infinities. That is 2ω+3 is no fun as a end answer and ω+1 is no fun as answer but 2ω+3ω+1=2ωω+1+3ω+1=2−y+3ϵ−z where y and z are smaller than standard so it has a usable standard part answer of 2.