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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers and let $\\alpha \u003e 0$ be a real number that is not an integer.\nLet $n = \\ceil{\\alpha}$.\nLet $f \\colon [a, b] \\to \\R$ have $n - 1$ continuous derivatives such that $f^{(n-1)}$ is absolutely continuous, so that $f^{(n)}$ exists almost everywhere and is integrable.\nThe *Caputo fractional derivative* of order $\\alpha$ of $f$ is\n$${}^{C}D_a^\\alpha f(x) = \\frac{1}{\\Gamma(n - \\alpha)} \\int_a^x (x - t)^{n - \\alpha - 1} f^{(n)}(t) \\, dt, \\qquad a \u003c x \\le b,$$\nthat is, the [[Riemann–Liouville integral|JTKCZ44NN3]] $I_a^{n - \\alpha}$ applied to the [[higher derivative|04TKTF8KDK]] $f^{(n)}$.\nHere $\\Gamma$ is the [[gamma function|0H5SYBSED9]].\nFor an integer $\\alpha = n$, one sets ${}^{C}D_a^n f = f^{(n)}$.\n\n## Remarks\n\nExample: let $a = 0$, $\\alpha = \\tfrac{1}{2}$, so $n = 1$, and $f(x) = x$.\nThen $f' = 1$ and ${}^{C}D_0^{1/2} f(x) = \\frac{1}{\\Gamma(1/2)} \\int_0^x (x - t)^{-1/2} \\, dt = \\frac{2\\sqrt{x}}{\\sqrt{\\pi}}$.\nFor a constant $f = c$, $f^{(n)} = 0$ and the Caputo derivative is $0$.\nNon-example: the Riemann–Liouville derivative $\\frac{d}{dx} I_0^{1/2} 1 = \\frac{1}{\\sqrt{\\pi x}}$ is not $0$, so it does not agree with the Caputo derivative on constants.\nIn general, under the hypotheses above, the Caputo derivative equals the Riemann–Liouville derivative of $f(x) - \\sum_{k=0}^{n-1} \\frac{f^{(k)}(a)}{k!} (x - a)^k$.\n\n## Explanations\n\n### L1 human\n\nThe Caputo derivative takes an ordinary derivative of whole order first, and then smooths with a fractional integral.\nThis keeps the familiar rule that constants have derivative zero.\n",
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