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  "body": "\n## Statement\n\nLet $f(x) = \\sum_{n=0}^{\\infty} a_n (x - c)^n$ be a [[power series|0VP56B4EMX]] with [[radius of convergence|0660QX1VPR]] $R \u003e 0$.\nThen $f$ is differentiable on $(c - R, c + R)$, and\n$$f'(x) = \\sum_{n=1}^{\\infty} n a_n (x - c)^{n-1}.$$\nThe derived series also has radius of convergence $R$.\n\n## Remarks\n\nThe [[Cauchy–Hadamard formula|07E5MB1CA2]] gives the radius of the derived series, since $\\sqrt[n]{n} \\to 1$. Then [[term-by-term differentiation|0TJ8QBJX5K]] applies on each $[c - r, c + r]$ with $r \u003c R$.\nBy induction, $f$ is infinitely differentiable, and $a_n = f^{(n)}(c)/n!$. So the series is the Taylor series of $f$ at $c$.\nMathlib states differentiability as `HasFPowerSeriesOnBall.differentiableOn`.\n\n## Explanations\n\n### L1 human\n\nA power series can be differentiated like a polynomial, one term at a time.\nThe new series converges on the same open interval.\n",
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