Chain Rule
y = d/dx (sin(x² − t))
Open in the app The dials and keys named below are the app's.
What it draws
The slope of a chirp. The wave underneath is sin(x² − t), whose crossings fall where x² is a multiple of π, at 1.77, 2.51, 3.07, 3.54 and so on. They crowd together further out, because squaring makes the phase run faster the larger x becomes.
What is plotted is not that wave but its derivative, and the derivative is a different shape: 2x · cos(x² − t). It keeps the crowding, since the cosine turns over exactly where the sine did, and it gains something the original never had. The height grows.
Where the height comes from
That growth is the chain rule, drawn. Differentiating sin(f) gives cos(f) multiplied by the slope of f itself, and here f = x² − t, whose slope is 2x. So the wave is bounded between −1 and 1 while its slope is bounded by ±2x: at x = 3 the envelope is 6, and at x = 5 it is 10.
It makes sense from the picture alone. Out at the edges the wave completes a whole turn in a very short distance, so it must climb and fall steeply to do it. Near the origin it has all the room it needs and barely leans at all.
Try
- Look at the wave itself with sin(x² − t). Same crowding, flat envelope, and the contrast between the two is the whole idea.
- Make the inside linear with d/dx (sin(3x − t)). The slope is now a constant 3, so the envelope is flat: the growth came from the square, not from differentiating.
- Cube it instead, d/dx (sin(x³ − t)). The inner slope is 3x², so the envelope curves upward rather than climbing straight.
- Take the slope of the slope, d/dx (d/dx (sin(x² − t))). It works, and it is visibly rougher: a difference of differences keeps about half the digits.