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  "body": "\n## Statement\n\nThe [[natural logarithm|0044CPHJBV]] is differentiable on $(0, \\infty)$, and\n$$\\frac{d}{dx} \\ln x = \\frac{1}{x} \\quad \\text{for } x \u003e 0.$$\n\n## Remarks\n\nBy the [[inverse function rule|0M0RZ3HSB0]] and [[$\\exp' = \\exp$|0EX9GFJ0B3]], the derivative at $x = e^y$ is $\\frac{1}{e^y} = \\frac{1}{x}$.\nFor $x \\ne 0$, the derivative of $\\ln |x|$ is $1/x$.\nBy the chain rule, $(\\ln f(x))' = f'(x)/f(x)$ when $f \u003e 0$. Mathlib states the main claim as `Real.hasDerivAt_log`.\n\n## Explanations\n\n### L1 human\n\nThe slope of the graph of $\\ln$ at $x$ is $1/x$.\nSo the logarithm grows more and more slowly as $x$ gets large.\n",
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