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  "body": "\n## Summary\n\nThe Schwarzian derivative of a locally injective holomorphic function $f$ is the expression $f'''/f' - \\frac{3}{2}(f''/f')^2$.\nIt vanishes exactly for Möbius transformations and does not change when $f$ is composed on the left with a Möbius transformation.\nSo it measures how far $f$ is from being a Möbius map.\nIt appears in univalence criteria, in the theory of linear second order differential equations, and in Teichmüller theory.\n\n## Explanations\n\n### L1 human\n\nThe ordinary derivative is zero for constant functions, and the second derivative is zero for linear functions.\nThe Schwarzian derivative is a third order expression that is zero for all Möbius maps $z \\mapsto (az + b)/(cz + d)$.\nIt ignores any Möbius map applied after $f$, so it only sees the part of $f$ that a Möbius map cannot explain.\nFor the exponential function it is the constant $-1/2$.\n",
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