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Why are the derivatives of periodic functions periodic?

Answer 1

Scarlett Allison

See the explanation section below.

#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)#

Apply the definition of derivative. (Alternatively, but not inevitably, apply the chain rule.)

#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.

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Answer 2

Christopher Allers

The derivatives of periodic functions are periodic because the rate of change of the function repeats itself over regular intervals, corresponding to the period of the function. Since the derivative measures the rate of change of the function, if the function itself repeats its pattern periodically, the rate of change (derivative) will also repeat in the same manner. This periodicity in derivatives is a result of the underlying periodic behavior of the original function.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.