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  "body": "\n## Statement\n\nLet $V$ and $W$ be normed vector spaces over $\\mathbb{R}$, let $U \\subseteq V$ be open, let $f, g \\colon U \\to W$, let $x \\in U$, and let $c \\in \\mathbb{R}$.\nSuppose that $f$ and $g$ are Fréchet differentiable at $x$, with [[Fréchet derivatives|08EVNYZF42]] $Df(x)$ and $Dg(x)$.\nThen $f + g$ and $c f$ are Fréchet differentiable at $x$, and\n$$D(f + g)(x) = Df(x) + Dg(x), \\qquad D(c f)(x) = c \\, Df(x).$$\n\n## Remarks\n\nThe proof adds the two remainders: a sum of two terms that are $o(\\|h\\|)$ is again $o(\\|h\\|)$.\nSo the maps that are differentiable at $x$ form a vector space, and $f \\mapsto Df(x)$ is a linear map on it.\nThe same rules hold for the [[partial derivatives|09EGC6F8PT]] and for the [[directional derivatives|06KWGHDN2J]].\n\n## Explanations\n\n### L1 human\n\nThe derivative of a sum is the sum of the derivatives, and constants come out, just as in one variable.\nFor example, $D(x^2 + 3 y)$ at a point $(a, b)$ is the linear map $h \\mapsto 2 a h_1 + 3 h_2$, the sum of the derivatives of $x^2$ and of $3 y$.\n",
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