In non-standard analysis the following proof of the product rule is considered valid:
y = uv
Δy = (u + Δu)(v + Δv) - uv
= uΔv + vΔu + ΔuΔv
Δy/Δx = uΔv/Δx + vΔu/Δx + ΔuΔv/Δx
dy/dx = st(uΔv/Δx) + st(vΔu/Δx) + st(ΔuΔv/Δx)
= u dv/dx + v du/dx + 0 * st(Δv/Δx)
= u dv/dx + v du/dx
I would like to understand why st(ΔuΔv/Δx) = 0 without resorting to limits. What I am not understanding is how one can apply the "Increment Theorem" (which states Δy = f'(x) + 𝜀Δx for some infinitesimal 𝜀) without knowing before hand what f'(x) is. How can we even know that ΔuΔv/Δx is infinitesimal without looking to the limit, and even if we knew this how would we know if we had the correct 𝜀Δx?