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  "body": "\n## Statement\n\nLet $g$ be a real function on an open interval $I$, differentiable at $a \\in I$, with $g(a) \\ne 0$.\nThen $1/g$ is differentiable at $a$, and\n$$\\left( \\frac{1}{g} \\right)'(a) = -\\frac{g'(a)}{g(a)^2}.$$\n\n## Remarks\n\nThe difference quotient is $-\\frac{g(a + h) - g(a)}{h} \\cdot \\frac{1}{g(a + h) g(a)}$. Continuity of $g$ at $a$ gives the limit.\nFor example, $(1/x)' = -1/x^2$ for $x \\ne 0$.\nMathlib states it as `HasDerivAt.inv`.\n\n## Explanations\n\n### L1 human\n\nWhen $g$ grows, $1/g$ shrinks, so the two derivatives have opposite signs.\nThe factor $1/g^2$ shows that $1/g$ changes fast where $g$ is small.\n",
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