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  "body": "\n## Statement\n\nLet $I$ be an open interval and let $f \\colon I \\to \\mathbb{R}$ be continuous and strictly monotone.\nAssume that $f$ is differentiable at $a \\in I$ with $f'(a) \\ne 0$, and let $b = f(a)$.\nThen $f^{-1}$ is differentiable at $b$, and\n$$(f^{-1})'(b) = \\frac{1}{f'(a)} = \\frac{1}{f'(f^{-1}(b))}.$$\n\n## Remarks\n\nThe inverse is [[continuous|07BY47HG5S]], so $f^{-1}(b + k) \\to a$ as $k \\to 0$. The difference quotient of $f^{-1}$ at $b$ is the reciprocal of a difference quotient of $f$ at $a$.\nIf $f'(a) = 0$, then $f^{-1}$ is not differentiable at $b$. An example is $f(x) = x^3$ at $a = 0$.\nFor example, it gives the [[derivative of the logarithm|07X1QA6BRW]] from the derivative of the exponential function.\n\n## Explanations\n\n### L1 human\n\nThe graph of $f^{-1}$ is the graph of $f$ reflected in the line $y = x$.\nThe reflection turns a slope $m$ into the slope $1/m$.\n",
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