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  "body": "\n## Statement\n\nLet $V$, $W_1$ and $W_2$ be normed vector spaces over $\\mathbb{R}$, let $U \\subseteq V$ be open, and let $x \\in U$.\nLet $f_1 \\colon U \\to W_1$ and $f_2 \\colon U \\to W_2$ be Fréchet differentiable at $x$.\nThen the map $F \\colon U \\to W_1 \\times W_2$, $F(y) = (f_1(y), f_2(y))$, is Fréchet differentiable at $x$, and\n$$DF(x)(h) = \\big(Df_1(x)(h), \\, Df_2(x)(h)\\big) \\quad \\text{for each } h \\in V.$$\nHere $W_1 \\times W_2$ has the norm $\\|(w_1, w_2)\\| = \\max(\\|w_1\\|, \\|w_2\\|)$.\n\n## Remarks\n\nThe converse also holds: if $F$ is differentiable at $x$, then so are its two components, because each projection $W_1 \\times W_2 \\to W_i$ is a bounded linear map.\nAny norm on $W_1 \\times W_2$ that is equivalent to the max norm gives the same derivative.\nApplied $m$ times with $W_i = \\mathbb{R}$, this gives the [[derivative of a map into R^n by components|0NS0X75RS1]].\n\n## Explanations\n\n### L1 human\n\nTo differentiate a map with two outputs, differentiate each output and put the two results side by side.\nThe remainder of the pair is small exactly when both remainders are small.\n",
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