If you take the derivative of $\sin^2(x)$ and remember your double-angle formulas, you see that $$ \frac{\operatorname{d}}{\operatorname{d}\!x}\; \sin^2(x) = 2\sin(x)\cos(x) = \sin(2x)\,. $$ This looks surprisingly clean. You can say the rate of change of $\sin^2$ at a value is given by the $\sin$ of twice that value. Is this just a happy accident? Or is there some nice geometric/trigonometric intuition behind this that I'm not seeing?
2026-04-06 11:37:41.1775475461
Geometric intuition behind the derivative of $\sin^2(x)$ being $\sin(2x)$.
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