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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be open and let $f \\colon U \\to \\mathbb{R}^m$.\nFor an integer $k \\ge 0$, let $L^k(\\mathbb{R}^n; \\mathbb{R}^m)$ be the space of $k$-linear maps from $(\\mathbb{R}^n)^k$ to $\\mathbb{R}^m$, with $L^0(\\mathbb{R}^n; \\mathbb{R}^m) = \\mathbb{R}^m$.\nThe iterated derivatives are defined by recursion. First, $D^0 f = f$.\nSuppose that $D^k f \\colon U \\to L^k(\\mathbb{R}^n; \\mathbb{R}^m)$ is defined on $U$ and is [[differentiable on U|08S7FDRS5Z]].\nThen $D^{k+1} f(a)$, for $a \\in U$, is the $(k+1)$-linear map\n$$D^{k+1} f(a)(v_1, v_2, \\dots, v_{k+1}) = \\big(D(D^k f)(a)(v_1)\\big)(v_2, \\dots, v_{k+1}).$$\nHere $D(D^k f)(a)$ is the [[Fréchet derivative|08EVNYZF42]] of $D^k f$ at $a$, a linear map from $\\mathbb{R}^n$ to $L^k(\\mathbb{R}^n; \\mathbb{R}^m)$.\n\n## Remarks\n\nFor $k = 1$, $D^1 f = Df$. For $k = 2$, $D^2 f(a)(u, v)$ is the derivative of $x \\mapsto Df(x)(v)$ at $a$ in the direction $u$.\nOn basis vectors, $D^2 f(a)(e_i, e_j) = \\dfrac{\\partial}{\\partial x_i}\\left(\\dfrac{\\partial f}{\\partial x_j}\\right)(a)$. In general, $D^k f(a)(e_{i_1}, \\dots, e_{i_k})$ is a [[partial derivative of order k|02RSXVM1ZN]].\nFor a real-valued $f$, the matrix of the bilinear map $D^2 f(a)$ is the [[Hessian matrix|0693Y2METT]].\nIf $f$ is of class $C^k$, then $D^k f(a)$ is a symmetric $k$-linear map, by the [[symmetry of second derivatives|01V9FPBBVK]].\n\n## Explanations\n\n### L1 human\n\nThe first derivative takes one direction and gives a rate of change.\nThe second derivative takes two directions $u$ and $v$. It tells how the rate of change in direction $v$ itself changes as you move in direction $u$.\n",
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