{
  "aliases": [
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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ and $W \\subseteq \\mathbb{R}^m$ be open sets, let $g \\colon U \\to \\mathbb{R}^m$ with $g(U) \\subseteq W$, let $f \\colon W \\to \\mathbb{R}^k$, and let $a \\in U$.\nSuppose that $g$ is [[differentiable|0135XZZQ8M]] at $a$ and $f$ is differentiable at $g(a)$.\nThen the [[composition|0THBK9V825]] $f \\circ g \\colon U \\to \\mathbb{R}^k$ is differentiable at $a$, and\n$$D(f \\circ g)(a) = Df(g(a)) \\circ Dg(a).$$\n\n## Remarks\n\nIn matrix form, the [[Jacobian matrix|047NDTEPCV]] of $f \\circ g$ at $a$ is the product $J_f(g(a)) \\, J_g(a)$. In terms of partial derivatives, $\\dfrac{\\partial (f \\circ g)_i}{\\partial x_j}(a) = \\sum_{l=1}^m \\dfrac{\\partial f_i}{\\partial y_l}(g(a)) \\, \\dfrac{\\partial g_l}{\\partial x_j}(a)$.\nThe same rule, with the same proof, holds for [[Fréchet derivatives|08EVNYZF42]] of maps between normed spaces.\nA special case is the [[chain rule for a composition with a curve|0A8TJ0VHC7]].\n\n## Explanations\n\n### L1 human\n\nThe derivative of a composition is the composition of the derivatives: first the linear approximation of $g$, then the linear approximation of $f$.\nIn one variable this is $(f \\circ g)'(a) = f'(g(a)) \\, g'(a)$, a product of two numbers. In several variables the numbers become matrices.\n",
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        "locator": "General rule: Vector-valued multivariate functions",
        "url": "https://en.wikipedia.org/wiki/Chain_rule"
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        "url": "https://en.wikipedia.org/wiki/Fr%C3%A9chet_derivative"
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  "msc": [
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  "name": "Chain rule in several variables",
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