DERIVATIVE OF A PERIODIC FUNCTION

Question

Is the derivative of a periodic function periodic?

Answer

The statement is true, the derivative of a periodic function is periodic. Attached is the explanation and proof.

Answer

Is the derivative of a periodic function periodic ?
Suppose function f is differentiable periodic function with period T.
That is,  f(x) = f(x+T) for all x ∈ I .
Then,  f'(x) = lim(h->0) [ f(x+h) - f(x) ] / h =  lim(h->0) [ f(x+T+h) - f(x+T) ] / h  = f'(x+T)
OR   f'(x) = f'(x+T)
Hence, the derivative of a periodic function is always periodic.

Answer

The answer is actually "true"; at least, the derivative of any differentiable periodic functions is periodic. This is easy to prove:
Suppose f:R→Rf:R→R is a differentiable periodic function with period PP. Let g:R→Rg:R→R be the function g(x)=x+Pg(x)=x+P. Then f=f∘gf=f∘g so f′=(f∘g)′f′=(f∘g)′. By the chain rule, for all x∈Rx∈R we have
f′(x)=(f∘g)′(x)=f′(g(x))g′(x)=f′(x+P).f′(x)=(f∘g)′(x)=f′(g(x))g′(x)=f′(x+P).In other words, f′f′ is periodic with period PP.

Answer

Please find attachment. Read this well and use it to enhance as well as develop your understanding and knowledge about the given assignment. If you have any queries feel free to send me a message. Thanks
BEST OF LUCK :)
regards,Dr. Charles Kennedy (expertsolution)

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DERIVATIVE OF A PERIODIC FUNCTION

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