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  "body": "\n## Proof\n\nWe deduce the rule from the [[linearity of differentiation|0M0TKFRX2A]].\n\n1. Apply linearity to the functions $f$ and $g = f$, which are both differentiable at $a$, with $\\alpha = c$ and $\\beta = 0$.\n2. For each $x \\in I$, $(c f + 0 \\cdot f)(x) = c f(x)$. So $c f + 0 \\cdot f = c f$.\n3. Linearity gives that $c f$ is differentiable at $a$, with $(c f)'(a) = c f'(a) + 0 \\cdot f'(a) = c f'(a)$.\n\n## Idea\n\nA constant multiple is the special case of a linear combination where the second coefficient is zero. So the rule is a direct case of linearity.\n",
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