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  "body": "\n## Summary\n\nLet $f$ map an open subset of $\\mathbb{R}^n$ to $\\mathbb{R}^m$. The total derivative of $f$ at a point $a$ is the linear map $Df(a) \\colon \\mathbb{R}^n \\to \\mathbb{R}^m$ that best approximates $f$ near $a$. That is, $f(a + h) = f(a) + Df(a)(h) + o(\\|h\\|)$ as $h \\to 0$.\nIt is the [[Fréchet derivative|06EH97V2A7]] in the case of [[Euclidean spaces|0XPR1E85W0]].\nIts matrix in the standard bases is the [[Jacobian matrix|047NDTEPCV]] of $f$ at $a$.\n\n## Remarks\n\nThe total derivative accounts for changes in all variables at once. A partial derivative accounts for a change in one variable only.\nWhen $f$ depends on $t$ both directly and through other functions of $t$, the total derivative $\\dfrac{d}{dt} f(t, x(t))$ is the sum $\\dfrac{\\partial f}{\\partial t} + \\dfrac{\\partial f}{\\partial x} \\dfrac{dx}{dt}$, by the [[chain rule|0EAAS2EHDR]].\n\n## Explanations\n\n### L1 human\n\nFor a function of one variable, the derivative at a point is a number: the slope of the tangent line.\nFor a function of several variables, the derivative at a point is a linear map: it tells how the output changes for each small step in any direction.\n",
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