Summary: Trig derivatives from geometry

The power rule handled powers of t, but sin and cos are not powers of anything, so they need their own picture: a point moving around the unit circle at unit speed. From that single image both trig derivatives fall out at once, d/dx(sin x) = cos x and d/dx(cos x) = -sin x, and you can see exactly why the minus sign lands on cosine. No memorization, same nudge-and-look method as the power rule, different shape. This is the scan-it-in-five-minutes version.

Core ideas

• Sine and cosine are coordinates. A point on the unit circle at angle x sits at (cos x, sin x): cosine is the horizontal coordinate, sine the vertical. Differentiating them means finding how fast those coordinates change as the point moves, the point’s velocity.
• The point moves at unit speed. A radian is defined so arc length equals angle, so nudging the angle by dx moves the point arc length dx. That unit speed is the engine of the whole derivation (and the real reason calculus uses radians).
• Velocity is the position rotated a quarter turn. Circular motion is perpendicular to the radius, length 1, 90° counterclockwise. Rotating (a, b) by 90° CCW gives (-b, a), so the position (cos x, sin x) gives velocity (-sin x, cos x).
• Read off both derivatives. The velocity components are the coordinates’ rates: horizontal cos x changes at -sin x (so d/dx(cos x) = -sin x), vertical sin x changes at cos x (so d/dx(sin x) = cos x). The minus is on cosine because its coordinate shrinks as the point climbs counterclockwise.
• The formulas match the curves. At x = 0, sine climbs through the origin at slope cos 0 = 1 and cosine is flat at its peak (slope -sin 0 = 0). At x = π/2, sine is flat at its peak and cosine descends at slope -1. The derivatives describe slopes you can already see.
• Two payoffs. Near zero, sin(x) ≈ x (slope 1 at the origin, the first sliver of Taylor series). And differentiating sine twice gives -sin, the equation f'' = -f that makes sine and cosine the universal shape of oscillation (springs, pendulums, sound, AC, light).

What changes for you

Sine and cosine stop being a memorization burden (“which one gets the minus?”) and become one more thing you read off a picture, a point circling at unit speed, so you can always reconstruct the pair and never scramble the sign. The same nudge-and-look method that gave the power rule gives these too; only the shape changed. It also explains a convention that usually feels arbitrary: radians exist precisely to make the circling point move at unit speed, keeping every trig derivative free of a stray π/180. In machine learning these derivatives appear wherever oscillation or rotation does: transformer positional encodings (Vaswani et al., 2017) built from sine and cosine waves, 3D rotation matrices in differentiable rendering and pose estimation, and Fourier analysis in signal processing. The next lesson returns to combining functions, with the product rule and the chain rule.