This is a visualization of the product rule in calculus, which says that, for differentiable functions u and v,[br][br]d(u v) = v du + u dv[br][br][br]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, d(u v)</i> can be visualized as a change in area from a u*v rectangle to a rectangle that's slightly bigger by du in the u direction and by dv 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*du rectangle, and one a u*dv rectangle.

The curve above is parametric, meaning that each point on the curve represents (u(t), v(t)) for some t.[br][br]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 d(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.</p>[br][br]Now look at the red and blue rectangles. The area of the red rectangle is u*dv; similarly, the area of the blue rectangle is v*du 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[br][br]d(u v) is close to v*du+u*dv[br][br]Or, in the limiting case,[br][br]d(u v) = v*du+u*dv