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  "body": "\n## Statement\n\nLet $f$ and $g$ be real functions on an open interval $I$, both differentiable at $a \\in I$.\nThen $f g$ is differentiable at $a$, and\n$$(f g)'(a) = f'(a) g(a) + f(a) g'(a).$$\n\n## Remarks\n\nWrite $f(a + h) g(a + h) - f(a) g(a) = (f(a + h) - f(a)) g(a + h) + f(a) (g(a + h) - g(a))$. Divide by $h$ and use the [[continuity of $g$ at $a$|07ZMX4ANJS]].\nFor example, $(x \\sin x)' = \\sin x + x \\cos x$.\nMathlib states it as `HasDerivAt.mul`.\n\n## History\n\nThe discovery of the rule is credited to Gottfried Leibniz.\n\n## Explanations\n\n### L1 human\n\nThink of $f g$ as the area of a rectangle with sides $f$ and $g$.\nWhen both sides grow a little, the area grows by $f' g + f g'$, up to a tiny corner piece.\n",
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