Question

Why are the derivatives of periodic functions periodic?

Answer 1

See the explanation section below.

Explanation:

`f` is periodic if and only if there is an integer `c` such that `f(x+c) = f(x)` for all `x` in `"Dom"(f)`

If `f` is periodic for some `c` as above and `f'(x)` exists, then `f'(x+c)` exists and `f'(x+c) = f'(x)`

Use the definition of derivative. (You could use the chain rule instead, but it is not necessary.)

`f'(x+c) = lim_(hrarr0)(f((x+h)+c)-f(x+c))/h` `" "` (def of derivative)

`= lim_(hrarr0)(f((x+h))-f(x))/h` `" "` (periodicity of `f`)

`= f'(x)` `" "` `" "` (def of derivative)

Therefore, `f'` is also periodic.