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  "body": "\n## Statement\n\nThe cosine function is differentiable on $\\mathbb{R}$, and $\\cos'(x) = -\\sin x$ for all real $x$.\n\n## Remarks\n\n[[Differentiate the power series|002X7EK91H]] term by term. The derivative of $\\frac{(-1)^n x^{2n}}{(2n)!}$ is $-\\frac{(-1)^{n-1} x^{2n-1}}{(2n-1)!}$ for $n \\ge 1$.\nSo $\\sin'' = -\\sin$ and $\\cos'' = -\\cos$. Both functions solve $f'' + f = 0$.\nMathlib states it as `Real.hasDerivAt_cos`.\n\n## Explanations\n\n### L1 human\n\nThe slope of the cosine curve at $x$ is $-\\sin x$.\nThe cosine curve falls where the sine is positive.\n",
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