Why d(uv) ≠ u'v': Visualizing Calculus Product Rule Geometry

💡 Drag canvas edge or adjust sliders to alter x and Δx

Interactive Parameters Preset: x² · 3x

Evaluation Point x 2.00

Limit Step Δx 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

Why the product rule adds two growing strips

Your browser does not support video.

Read the explanation

The default preset multiplies x squared by three x. At x equals two, the sides are four and six, giving area twenty four. The side derivatives are four and three. The product rule multiplies each derivative by the other original side: four times six plus four times three equals thirty six. Multiplying the derivatives alone gives twelve and misses one entire side contribution. Increase x from two to two point two. The first side grows from four to four point eight four, a change of zero point eight four. The second grows from six to six point six, a change of zero point six. The added area has two strips: six times zero point eight four equals five point zero four, and four times zero point six equals two point four. The corner contributes zero point five zero four. Their sum is seven point nine four four, exactly the new area minus twenty four. The finite difference divides added area by the x step. At step zero point two, seven point nine four four divided by zero point two gives thirty nine point seven two. At step zero point zero two, the forward rate is thirty six point three six one two, much nearer the derivative thirty six. The shrinking corner is part of why the finite area picture approaches the product rule. A finite step is not exactly the derivative; this tool displays both quantities.