The period of an oscillation is the time it takes for a repeating cycle. For example, if a pendulum swings back and forth once each second, its period is one second. If the pendulum swings back and forth twice each second, its period is 0.5 seconds.

For a function f(x), we can think of its derivative as being an oscillation at some frequency ω (omega). This frequency is called the “amplitude” of the oscillation, because it determines how big the oscillations are. (In other words, it determines how far they move up or down.) So if f(x) has period T, then its derivative must also have period T. In fact, I can prove this by showing that every point on a periodic graph corresponds to exactly one point on a different periodic graph with the same frequency and amplitude.

If a function is periodic, with period c, then so is its derivative.

In other words, if f is periodic with period c, then df/dx = 0 for all x in [a, b] and f(x + c) = f(x) for all x in [a, b].

This can be proved by using integration by parts.

To do this, we’ll assume that f(x) is periodic with period c. Then we can define ƒ'(x) as follows: ƒ'(x) = ƒ(x + c) – ƒ(x). Now we have an integral that we can evaluate. By taking the limit as c approaches 0 (so that the integral becomes an ordinary derivative), we find that ƒ'(x) = 0 for all x in [a, b].

Last modified: August 16, 2022