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  "body": "\n## Statement\n\nThe arctangent is differentiable on $\\mathbb{R}$, and\n$$\\frac{d}{dx} \\arctan x = \\frac{1}{1 + x^2} \\quad \\text{for every real } x.$$\n\n## Remarks\n\nLet $y = \\arctan x$. By the [[inverse function rule|0M0RZ3HSB0]] and [[$\\tan' = 1 + \\tan^2$|0SAMR34WXC]], the derivative is $\\frac{1}{1 + \\tan^2 y} = \\frac{1}{1 + x^2}$.\nSo $\\int_0^1 \\frac{dx}{1 + x^2} = \\arctan 1 = \\frac{\\pi}{4}$.\nMathlib states it as `Real.hasDerivAt_arctan`.\n\n## Explanations\n\n### L1 human\n\nThe derivative of the arctangent is the simple rational function $1/(1 + x^2)$.\nThis connects the number $\\pi$ with integrals of rational functions.\n",
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