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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be an open set, let $f \\colon U \\to \\mathbb{R}$, let $a \\in U$, and let $i \\in \\{1, \\dots, n\\}$.\nLet $e_i$ be the $i$-th standard basis vector of [[Euclidean space|0XPR1E85W0]] $\\mathbb{R}^n$.\nThe partial derivative of $f$ with respect to $x_i$ at $a$ is the limit\n$$\\frac{\\partial f}{\\partial x_i}(a) = \\lim_{h \\to 0} \\frac{f(a + h e_i) - f(a)}{h},$$\nwhen this limit exists in $\\mathbb{R}$.\nOther notations for it are $\\partial_i f(a)$, $D_i f(a)$ and $f_{x_i}(a)$.\n\n## Remarks\n\nThe partial derivative is the ordinary derivative at $t = a_i$ of the function $t \\mapsto f(a_1, \\dots, a_{i-1}, t, a_{i+1}, \\dots, a_n)$ of one variable.\nSo you compute it with the rules of one-variable calculus, and you treat the other variables as constants.\nIt is the [[directional derivative|06KWGHDN2J]] in the direction $e_i$.\nThe existence of all partial derivatives at $a$ does not imply that $f$ is continuous at $a$, see [[this counterexample|0TNZC55X3J]].\n\n## Explanations\n\n### L1 human\n\nTo find $\\partial f / \\partial x$, pretend that $y$ is a fixed number and differentiate in $x$ as usual.\nFor $f(x, y) = x^2 y + \\sin y$, this gives $\\partial f / \\partial x = 2xy$, because $\\sin y$ is a constant for this purpose.\nGeometrically, it is the slope of the graph of $f$ when you walk in the $x$-direction.\n",
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