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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{C}$ be an open set and let $f : U \\to \\mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.\nThen the complex derivative $f'$ is holomorphic on $U$.\nThus, for each $n \\ge 1$, the $n$-th complex derivative $f^{(n)}$ exists and is holomorphic on $U$.\nAs a function of two real variables $(x, y)$ with $z = x + iy$, $f$ has continuous partial derivatives of all orders on $U$.\n\n## Remarks\n\nProof: near each point, $f$ is the sum of a power series by [[holomorphic functions are analytic|0826SGV3R0]]. A [[power series can be differentiated term by term|0YMP521ZR5]], and the result is again a power series.\nThere is no real analogue: $x \\mapsto x |x|$ is differentiable on $\\mathbb{R}$, but its derivative $2|x|$ is not differentiable at $0$.\nThe [[formula for the derivatives|0F0H5P5DED]] gives each $f^{(n)}$ as a circle integral.\n\n## Explanations\n\n### L1 human\n\nOnce a complex function has one derivative on an open set, it has a second one, a third one, and so on forever.\nSo in complex analysis there are no functions that are differentiable only once.\n",
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