# 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

Or, in the limiting case,Δ(u v) ≈ v Δu + u Δv

d(u v) = v du + u dv

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