# 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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