The derivative of a periodic function with period T is not a constant function, a periodic function with period T, or a non-periodic function. The correct answer is option 'D', which means none of the above.

A periodic function is a function that repeats its values in regular intervals called periods. In other words, if a function f(x) has a period T, then for any value of x, the function value at x+T is the same as the function value at x. Mathematically, this can be represented as f(x+T) = f(x).

Here's why the derivative of a periodic function is not any of the given options:

1. Constant function:
A constant function has the same value for all inputs. If the derivative of a periodic function were a constant function, it would imply that the rate of change of the function is constant throughout its interval. However, a periodic function typically has varying rates of change as it repeats its pattern. Therefore, the derivative cannot be a constant function.

2. Periodic function with period T:
If the derivative of a periodic function were a periodic function with the same period T, it would mean that the rate of change of the function repeats itself exactly every T units. While this might be true for simple periodic functions like sine or cosine, it is not generally true for all periodic functions. The derivative of a periodic function can have a different period or even be non-periodic.

3. Non-periodic function:
A non-periodic function does not repeat its values in regular intervals. If the derivative of a periodic function were a non-periodic function, it would mean that the rate of change of the function does not exhibit any regular pattern or repetition. However, periodic functions often exhibit regular patterns and repetitions, so their derivatives are unlikely to be non-periodic functions.

In conclusion, the derivative of a periodic function does not fall into any of the given options. It may or may not be periodic, depending on the specific characteristics of the periodic function.