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  "body": "\n## Definition\n\nLet $f$ be a real function defined on an interval $[a, a + r)$ with $r \u003e 0$.\nThe right derivative of $f$ at $a$ is the [[one-sided limit|0909PWZR62]]\n$$f'_+(a) = \\lim_{h \\to 0^+} \\frac{f(a + h) - f(a)}{h},$$\nwhen it exists.\nThe left derivative $f'_-(a)$ is defined in the same way, for $f$ on $(a - r, a]$ and $h \\to 0^-$.\n\n## Remarks\n\nA function on an open interval is differentiable at $a$ if and only if both one-sided derivatives exist and are equal. Their common value is the [[derivative|0F1WR9C83K]].\nOne-sided derivatives give a meaning to derivatives at the endpoints of a closed interval.\nFor $|x|$ at $0$, the right derivative is $1$ and the left derivative is $-1$.\nMathlib expresses them with `HasDerivWithinAt` on a half line.\n\n## Explanations\n\n### L1 human\n\nA one-sided derivative only looks at one side of the point.\nAt a corner of a graph, the slopes from the left and from the right exist but differ.\n",
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