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Sine wave
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=== Differentiation === [[Derivative|Differentiating]] any sinusoid with respect to time can be viewed as multiplying its amplitude by its angular frequency and advancing it by a quarter cycle: <math>\begin{align} \frac{d}{dt} [A\sin(\omega t + \varphi)] &= A \omega \cos(\omega t + \varphi) \\ &= A \omega \sin(\omega t + \varphi + \tfrac{\pi}{2}) \, . \end{align}</math> A [[differentiator]] has a [[Zeros and poles|zero]] at the origin of the [[complex frequency]] plane. The [[Gain (electronics)|gain]] of its [[frequency response]] increases at a rate of +20 [[Decibel|dB]] per [[Decade (log scale)|decade]] of frequency (for [[root-power]] quantities), the same positive slope as a 1{{Sup|st}} order [[high-pass filter]]'s [[stopband]], although a differentiator doesn't have a [[cutoff frequency]] or a flat [[passband]]. A n{{Sup|th}}-order high-pass filter approximately applies the n{{Sup|th}} time derivative of [[signals]] whose frequency band is significantly lower than the filter's cutoff frequency.
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