Product Rule

This is a visualization of the product rule in calculus, which says that, for differentiable functions u and v,

d(u v) = v du + u dv

The basic idea of this visualization is that any product a b can be visualized as the area of an a×b rectangle. Since we are visualizing differentials, Δ(u v) can be visualized as a change in area from a u×v rectangle to a rectangle that's slightly bigger by Δu in the u direction and by Δv in the v direction. We aim to see that this change in area is almost entirely accounted for by the area of two skinny rectangles, one a v×Δu rectangle, and one a u×Δv rectangle.

The curve above is a parametric curve of two differentiable functions, u(t) and v(t). Each point on the curve represents (u(t), v(t)) for some t.

The area of the rectangle bounded by (0, 0) and (u(t), v(t)) is u(t) v(t). Similarly, The area of the rectangle bounded by (0, 0) and (u(t+h), v(t+h)) is u(t+h) v(t+h). Thus the area of the colored area, the difference between the two rectangles, is Δ(u v). Dragging the h slider to the left makes that area shrink, approximating d(u v). Now drag the slider back to the right again.

Now look at the red and blue rectangles. The area of the red rectangle is u Δv; similarly, the area of the blue rectangle is v Δu. Now slide the h slider to the left and notice that the brown rectangle, the only part of the d(u v) shape not included in the red and blue rectangles, disappears to insignificance. Thus

Δ(u v) ≈ v Δu + u Δv
Or, in the limiting case,
d(u v) = v du + u dv

Jon Dreyer • Math tutorComputer Science
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