Yeah, the Fourier transform diagonalizes differentiation. Or in other words "differentiation is a Fourier multiplier". Integration is like a low-pass filter (you divide by the amplitude of a component by its frequency and so higher frequencies get quieter) and differentiation is "noisy" since high frequency components get louder.
d^2/dt^2(sin kt) = -k^2 sin(kt)
This idea is used in the study of partial differential equations, especially dispersive.