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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be an open set and let $f \\colon U \\to \\mathbb{R}^m$.\nThe function $f$ is differentiable on $U$ when it is [[differentiable|0135XZZQ8M]] at each point $a \\in U$.\nThen the derivative of $f$ is the map $Df \\colon U \\to L(\\mathbb{R}^n, \\mathbb{R}^m)$, $a \\mapsto Df(a)$, where $L(\\mathbb{R}^n, \\mathbb{R}^m)$ is the space of linear maps from $\\mathbb{R}^n$ to $\\mathbb{R}^m$.\n\n## Remarks\n\nThe function $f$ is continuously differentiable, or of class $C^1$, on $U$ when it is differentiable on $U$ and $Df$ is continuous. Here $L(\\mathbb{R}^n, \\mathbb{R}^m)$ carries the topology of $m \\times n$ matrices.\nFor a set $S$ that is not open, one uses derivatives within $S$ instead. On an open set the two notions agree.\nEach polynomial map $\\mathbb{R}^n \\to \\mathbb{R}^m$ is differentiable on $\\mathbb{R}^n$.\n\n## Explanations\n\n### L1 human\n\nDifferentiable on $U$ means that there is a good linear approximation at each point of $U$, not only at one point.\nThe approximation changes from point to point, so the derivative is itself a function on $U$.\n",
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