Math Proofs
Constant rule in calculus
The derivative of a constant is always $ 0 $.
$$ \tfrac{d}{dx}(c) = 0 $$
Define function $ f $.
$$ f(x) = c $$
Take the definition of the derivative.
$$ f'(x) = \lim_{h \to 0}\left(\frac{f(x + h) - f(x)}{h}\right) $$
Substitue $ f(x) = c $ and $ f(x + h) = c $.
$$ f'(x) = \lim_{h \to 0}\left(\frac{c - c}{h}\right) $$
Finally, simplify.
$$ f'(x) = \lim_{h \to 0}\left(\frac{0}{h}\right) $$
$$ f'(x) = \lim_{h \to 0}\big(0\big) $$
$$ f'(x) = 0 $$
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Printed on January 28, 2023