information:filtering, structural:reformatting
Smooth infinitesimal analysis

In the context of Lawvere's approach to infinitesimals, let d x {\displaystyle dx} be a nilsquare infinitesimal. Then d u = u ′ d x {\displaystyle du=u'\ dx} and d v = v ′ d x {\displaystyle dv=v'\ dx}, so that

since d u d v = u ′ v ′ (d x) 2 = 0. {\displaystyle du\,dv=u'v'(dx)^{2}=0.} Dividing by d x {\displaystyle dx} then gives d (u v) d x = u d v d x + v d u d x {\displaystyle {\frac {d(uv)}{dx}}=u{\frac {dv}{dx}}+v{\frac {du}{dx}}} or (u v) ′ = u ⋅ v ′ + v ⋅ u ′ {\displaystyle (uv)'=u\cdot v'+v\cdot u'}.