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  "body": "\n## Definition\n\nLet $V$ and $W$ be normed vector spaces over $\\mathbb{R}$, let $U \\subseteq V$ be an open set, let $f \\colon U \\to W$, and let $x \\in U$.\nA bounded linear map $A \\colon V \\to W$ is a Fréchet derivative of $f$ at $x$ when\n$$\\lim_{h \\to 0} \\frac{\\|f(x + h) - f(x) - A h\\|_W}{\\|h\\|_V} = 0.$$\nThe limit is over $h \\in V \\setminus \\{0\\}$ with $x + h \\in U$.\nWhen such an $A$ exists, $f$ is Fréchet differentiable at $x$, with the notation $Df(x) = A$ or $f'(x) = A$.\n\n## Remarks\n\nThe Fréchet derivative is unique when it exists, see the [[uniqueness of the derivative|0762WAE49P]].\nFor $V = W = \\mathbb{R}$, a linear map $A$ is multiplication by a number $a$. Then the condition says that $f$ has the derivative $a$ at $x$ in the usual sense.\nFor $V = \\mathbb{R}^n$ and $W = \\mathbb{R}^m$, each linear map is bounded, and $Df(x)$ is the [[total derivative|0FNKFT8H6E]] of $f$ at $x$. Its matrix is the [[Jacobian matrix|047NDTEPCV]].\nIf $f$ is Fréchet differentiable at $x$, then $f$ is continuous at $x$.\n\n## Explanations\n\n### L1 human\n\nNear $x$, the change $f(x + h) - f(x)$ is close to $A h$, a linear function of the step $h$.\nThe error is small even compared with the length of the step, when the step is short.\nSo $A$ plays the role of the slope of a tangent line, but now the tangent object is a linear map.\n",
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