Derivative of a·sin(bx) at a Point

The slope of the sine curve is the cosine curve: where sin x is climbing fastest, at x = 0, the cosine is at its peak of 1, and where the sine levels off at the top of its wave, the cosine passes through 0. Stretch the wave to a·sin(bx) and both numbers come out in front, a from the height and b from the chain rule, giving ab·cos(bx). Worked example: y = 3 sin(2x) at x = π/6 = 0.5236. Then bx = π/3, cos(π/3) = 0.5, and the slope is 3 × 2 × 0.5 = 3.

The classic mistake is working in degrees. The rule d/dx sin x = cos x is true only in radians, because only in radians does sin h / h approach 1 as h shrinks. Enter the same point as x = 30 degrees and cos(60°) is still 0.5, but the slope per degree is 3 × π/180 = 0.0524, about 57 times smaller than the per-radian answer (180/π = 57.3). Set the calculator to radians, or convert with radians = degrees × π/180 before you start. The other slip is dropping b, the chain-rule factor. This is the arithmetic of everything that oscillates: a mass on a spring at x = A sin(ωt) moves at Aω cos(ωt), fastest as it passes the middle, and an alternating voltage V₀ sin(ωt) changes fastest at its zero crossings, which is when the current into a capacitor across it peaks.