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  "body": "\n## Statement\n\nLet $n \\ge 1$ be a natural number.\nThen the function $x \\mapsto x^n$ is differentiable on $\\mathbb{R}$, and its derivative at $x$ is $n x^{n-1}$.\n\n## Remarks\n\nInduction on $n$ with the [[product rule|0V9A43H7WV]] gives $(x^{n+1})' = (x^n \\cdot x)' = n x^{n-1} \\cdot x + x^n = (n + 1) x^n$.\nThe same formula holds for negative integers $n$ when $x \\ne 0$, and for [[real exponents|0KY08TWDMQ]] when $x \u003e 0$.\nMathlib states it as `hasDerivAt_pow`.\n\n## Explanations\n\n### L1 human\n\nTo differentiate a power, bring the exponent down as a factor and lower the exponent by one.\nFor example, the derivative of $x^5$ is $5 x^4$.\n",
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