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  "body": "\n## Statement\n\nLet $f$ be a real function on an open interval $I$ that is differentiable at $a \\in I$.\nLet $g$ be a real function on an open interval $J$ with $f(I) \\subseteq J$, and assume that $g$ is differentiable at $f(a)$.\nThen $g \\circ f$ is differentiable at $a$, and\n$$(g \\circ f)'(a) = g'(f(a)) \\, f'(a).$$\n\n## Remarks\n\nA naive proof multiplies and divides by $f(a + h) - f(a)$. This fails when $f(a + h) = f(a)$ for small $h$. A correct proof uses the linear approximation of $g$ at $f(a)$.\nFor example, $(\\sin(x^2))' = 2 x \\cos(x^2)$.\nIn Leibniz notation, $\\frac{dz}{dx} = \\frac{dz}{dy} \\cdot \\frac{dy}{dx}$. Mathlib states the rule as `HasDerivAt.comp`.\n\n## History\n\nGottfried Wilhelm Leibniz seems to have used the rule first, in a memoir of 1676.\n\n## Explanations\n\n### L1 human\n\nSuppose $y$ changes $f'(a)$ times as fast as $x$, and $z$ changes $g'(y)$ times as fast as $y$.\nThen $z$ changes $g'(y) f'(a)$ times as fast as $x$.\nThe rates multiply.\n",
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