Derivatives, Products and Totatives

Calculus Level 3

Consider a function \(f:\mathbb{R}\mapsto \mathbb{R}\) defined as,

\[f(x)=\prod_{i=1}^{2014} \left(\frac{(x-i)}{(\sqrt{x}-\sqrt{i})(\sqrt{x}+\sqrt{i})+\varphi(2)}\right)\]

Compute the value of the following \((\textbf{if it exists})\):

\[f'\left(\sqrt{2014}\right)+2014\]


\(\textbf{Details and Assumptions:}\)

\(\bullet\quad \varphi(x)\) denotes the Euler's Totient function.

\(\bullet\quad \displaystyle \prod_{i=a}^b f(i)=f(a)f(a+1)\cdots f(b-1)f(b)\)

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