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Answer 1 · 2017-04-15T09:06:35

#f''(x)=-n^4 e^(i n^2x)#

By Euler's Formula: #cos nx+i sin nx=e^(i nx)#,

#f(x)=e^(ix)cdot e^(i 3x)cdot e^(i 5x)cdots e^(i (2n-1)x)=e^(i[1+3+5+cdots (2n-1)]x)#

By the sum formula: #1+3+5+cdots+(2n-1)=n^2#,

#f(x)=e^(i n^2x)#

By differentiating w.r.t. #x#,

#f'(x)=i n^2 e^(i n^2x)#

By differentiating w.r.t.#x# one more time,

#f''(x)=i^2 n^4 e^(i n^2x)=-n^4e^(i n^2x)#

I hope that this was clear.