Let denote a function differentiable for all such that and the Fourier Transforms (FT) of both and exist, where denotes the time derivative of . Then we have
where denotes the Fourier transform of . In operator notation:
Proof: This follows immediately from integration by parts:
The differentiation theorem is implicitly used in §E.6 to show that audio signals are perceptually equivalent to bandlimited signals which are infinitely differentiable for all time.