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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, let $f, g \\colon U \\to \\mathbb{R}$, and let $a \\in U$.\nSuppose that $f$ and $g$ are [[differentiable|0135XZZQ8M]] at $a$.\nThen the product $f g$ is differentiable at $a$, and its derivative is\n$$D(f g)(a) = f(a) \\, Dg(a) + g(a) \\, Df(a).$$\nThat is, $D(f g)(a)(h) = f(a) \\, Dg(a)(h) + g(a) \\, Df(a)(h)$ for each $h \\in \\mathbb{R}^n$.\n\n## Remarks\n\nIn terms of the [[gradient|0ABK9V9Y64]], the rule reads $\\nabla (f g) = f \\nabla g + g \\nabla f$.\nThe same rule holds for each partial derivative: $\\partial_i (f g) = f \\, \\partial_i g + g \\, \\partial_i f$.\nMore generally, let $B$ be a bounded bilinear map. If $f$ and $g$ are [[Fréchet differentiable|08EVNYZF42]] at $a$, then $D(B(f, g))(a)(h) = B(Df(a) h, g(a)) + B(f(a), Dg(a) h)$. This gives product rules for the dot product and the cross product.\n\n## Explanations\n\n### L1 human\n\nThis is the familiar rule $(f g)' = f' g + f g'$, applied to the change in each direction at once.\nFor $f = x$ and $g = y$ on $\\mathbb{R}^2$, the product $x y$ has the derivative $h \\mapsto y h_1 + x h_2$ at the point $(x, y)$.\n",
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      {
        "locator": "Banach space",
        "url": "https://en.wikipedia.org/wiki/Product_rule"
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        "locator": "In vector calculus",
        "url": "https://en.wikipedia.org/wiki/Product_rule"
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  "msc": [
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  "name": "Product rule in several variables",
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