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## Sinc function

, 13:07, 4 September 2011
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===First derivative===
''Computation at $x = 0$'':  {{point differentation incorrect}} Here, we need to compute the derivative using first principles, as a limit of a [[difference quotient]]:
$\operatorname{sinc}' 0 := \lim_{x \to 0} \frac{\operatorname{sinc} x - \operatorname{sinc} 0}{x - 0} = \lim_{x \to 0} \frac{(\sin x)/x - 1}{x} = \lim_{x \to 0} \frac{\sin x - x}{x^2}$
$\lim_{x \to 0} \frac{\sin x - x}{x^2} \stackrel{*}{=} \lim_{x \to 0} \frac{\cos x - 1}{2x} \stackrel{*}{=} \lim_{x \to 0} \frac{-\sin x}{2} = \frac{-\sin 0}{2} = 0$

{{point differentation incorrect}}
''Computation at $x \ne 0$'': At any such point, we know that the function looks like $(\sin x)/x$ in an [[open interval]] about the point, so we can use the [[quotient rule for differentiation]], which states that:
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