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  "body": "\n## Statement\n\nLet $a$ be a real number. The function $x \\mapsto x^a$ is differentiable on $(0, \\infty)$, and\n$$\\frac{d}{dx} x^a = a x^{a-1} \\quad \\text{for } x \u003e 0.$$\n\n## Remarks\n\nWrite $x^a = e^{a \\ln x}$. The [[chain rule|0RXX2PT1MQ]] with [[$\\exp' = \\exp$|0EX9GFJ0B3]] and [[$\\ln' x = 1/x$|07X1QA6BRW]] gives $e^{a \\ln x} \\cdot \\frac{a}{x} = a x^{a-1}$.\nFor example, $\\frac{d}{dx} \\sqrt{x} = \\frac{1}{2 \\sqrt{x}}$.\nSimilarly, $\\frac{d}{dx} a^x = a^x \\ln a$ for a fixed $a \u003e 0$. Mathlib states the main claim as `Real.hasDerivAt_rpow_const`.\n\n## Explanations\n\n### L1 human\n\nThe power rule for whole exponents also works for real exponents, if $x \u003e 0$.\nBring the exponent down and lower it by one.\n",
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