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  "body": "\n## Statement\n\nThe functions $\\sinh$, $\\cosh$ and $\\tanh$ are differentiable on $\\mathbb{R}$, and for every real $x$\n$$\\sinh'(x) = \\cosh x, \\qquad \\cosh'(x) = \\sinh x, \\qquad \\tanh'(x) = \\frac{1}{\\cosh^2 x} = 1 - \\tanh^2 x.$$\n\n## Remarks\n\nThe first two follow from [[$\\exp' = \\exp$|0EX9GFJ0B3]] and $(e^{-x})' = -e^{-x}$.\nThe third follows from the [[quotient rule|0BS0NJPKVE]] and the identity [[$\\cosh^2 - \\sinh^2 = 1$|06QCTRFBM0]].\nSo $\\sinh'' = \\sinh$ and $\\cosh'' = \\cosh$. Mathlib states the first one as `Real.hasDerivAt_sinh`.\n\n## Explanations\n\n### L1 human\n\nUnlike sine and cosine, there is no minus sign here.\nEach of $\\sinh$ and $\\cosh$ is the derivative of the other.\n",
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