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.

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

Golden Ratio

(\phi)

Napier's constant

(e)

2016 Undefined

\pi

2015 1729 2014

1 solution

Caleb Townsend
Mar 13, 2015

$\prod_{i=1}^{2014}\frac{x - i}{(\sqrt{x} - \sqrt{i})(\sqrt{x} + \sqrt{i}) + \varphi(2)} \\ = \prod_{i=1}^{2014}\frac{x - i}{x - i + 1} \\ = (\frac{x - 1}{x})(\frac{x - 2}{x - 1})... (\frac{x - 2014}{x - 2013}) \\ = \frac{x - 2014}{x} = 1 - \frac{2014}{x} = f(x)$

$f'(x) = \frac{2014}{x^2} \\ f'(\sqrt{2014}) + 2014 = \frac{2014}{2014} + 2014 \\ = \boxed{2015}$

Note that $f(x)=\dfrac{x-2014}{x}$ only for $x\in \mathbb{R}\setminus \{i\}_{i=0}^{i=2013}$ .

It'd be nice if you mention this little part in your solution to make it complete.

Prasun Biswas - 6 years, 3 months ago

Derivatives, Products and Totatives

1 solution

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