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  "body": "\n## Statement\n\nThe sine function is differentiable on $\\mathbb{R}$, and $\\sin'(x) = \\cos x$ for all real $x$.\n\n## Remarks\n\n[[Differentiate the power series|002X7EK91H]] term by term. The derivative of $\\frac{(-1)^n x^{2n+1}}{(2n+1)!}$ is $\\frac{(-1)^n x^{2n}}{(2n)!}$, the term of the cosine series.\nWith the addition formula, the claim reduces to the limits $\\frac{\\sin h}{h} \\to 1$ and $\\frac{\\cos h - 1}{h} \\to 0$.\nMathlib states it as `Real.hasDerivAt_sin`.\n\n## Explanations\n\n### L1 human\n\nThe slope of the sine curve at $x$ is $\\cos x$.\nAt $x = 0$ the sine curve rises with slope $1$, and at $x = \\pi/2$ it is flat.\n",
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