💡 Drag canvas edge or adjust sliders to alter x and Δx
Interactive Parameters
Preset: x² · 3x
2.00
0.200
Geometric Area Decomposition ΔA
Base Area (u·v)
12.00
Original rectangle
Right Strip (u·Δv)
2.40
Width u × height Δv
Top Strip (v·Δu)
2.64
Height v × width Δu
Corner (Δu·Δv)
0.264
Vanishes as Δx → 0
Derivative Comparison at x
Naive Assumption
u'(x) · v'(x)
12.00
True Product Rule
u'(x)v(x) + u(x)v'(x)
36.00
Why Δu·Δv vanishes: As Δx → 0, ΔA/Δx = u(Δv/Δx) + v(Δu/Δx) + Δu(Δv/Δx). The corner term contains an uncancelled factor Δu → 0, proving d(uv)/dx = u v' + v u'.
Observed Rate Delta: 0.00