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  "body": "\n## Definition\n\nLet $D \\subseteq \\R$ and let $f \\colon D \\to \\R$ be a [[function|0XHDTE5YMZ]].\nLet $x \\in D$ and let $h \\ne 0$ be a real number with $x + h \\in D$.\nThe *difference quotient* of $f$ at $x$ with increment $h$ is\n$$\\frac{f(x + h) - f(x)}{h}.$$\nEquivalently, for $a \\ne b$ in $D$, the difference quotient of $f$ between $a$ and $b$ is $\\dfrac{f(b) - f(a)}{b - a}$.\n\n## Remarks\n\nThe function $f$ has [[derivative|0F1WR9C83K]] $f'(x)$ at an interior point $x$ of $D$ exactly when the difference quotient has the [[limit|0TXTSEMHR5]] $f'(x)$ as $h \\to 0$.\nThe [[mean value theorem|0BARQY52N5]] says that, for $f$ continuous on $[a, b]$ and differentiable on $(a, b)$, the difference quotient between $a$ and $b$ equals $f'(c)$ for some $c \\in (a, b)$.\n\nExample: for $f(x) = x^2$, the difference quotient is $\\frac{(x + h)^2 - x^2}{h} = \\frac{2xh + h^2}{h} = 2x + h$.\nIt tends to $2x$ as $h \\to 0$, which is $f'(x)$.\n\nNon-example: the increment $h = 0$ is not allowed, since the expression $\\frac{f(x) - f(x)}{0}$ has no meaning.\nThis is why the derivative is a limit and not a value of the quotient.\n\nIn numerical analysis, the quotient is also called the forward difference quotient, and $\\frac{f(x + h) - f(x - h)}{2h}$ is the central difference quotient.\nThe symbol $\\frac{\\Delta y}{\\Delta x}$ is common notation for it.\n",
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