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  "body": "\n## Statement\n\nThe arcsine is differentiable on $(-1, 1)$, and\n$$\\frac{d}{dx} \\arcsin x = \\frac{1}{\\sqrt{1 - x^2}} \\quad \\text{for } -1 \u003c x \u003c 1.$$\n\n## Remarks\n\nLet $y = \\arcsin x$. The [[inverse function rule|0M0RZ3HSB0]] gives the derivative $\\frac{1}{\\cos y}$. By the [[Pythagorean identity|03H7WZ4T3K]], $\\cos y = \\sqrt{1 - x^2}$, since $\\cos y \u003e 0$ for $|y| \u003c \\pi/2$.\nThe arcsine is not differentiable at $\\pm 1$. Its graph has vertical tangents there.\nSimilarly, $\\frac{d}{dx} \\arccos x = -\\frac{1}{\\sqrt{1 - x^2}}$. Mathlib states the main claim as `Real.hasDerivAt_arcsin`.\n\n## Explanations\n\n### L1 human\n\nThe derivative of the arcsine is an algebraic function, although the arcsine itself is not.\nIt becomes infinite at the ends of the interval $[-1, 1]$.\n",
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