If f(x) = | cases x & x & 1 2 x & x ^ 2 & 2x x & x & 1 cases | ,...

Given f\left(x\right) = \left|\begin{cases}\cos x & x & 1 \\ 2\sin x & {x}^{2} & 2x \\ \tan x & x & 1\end{cases}\right|

$= \cos x\left({x}^{2} - 2{x}^{2}\right) - 2\sin x\left(x - x\right) + \tan x\left(2{x}^{2} - {x}^{2}\right)$

$= - {x}^{2}\cos x + \tan x . {x}^{2}$

$= {x}^{2}\left(\tan x - \cos x\right)$

$⇒ \lim_{x \to 0}\dfrac{{f}^{'}\left(x\right)}{x} = \lim_{x \to 0}\dfrac{2x\left(\tan x - \cos x\right) + {x}^{2}\left(\sec^{2}x + \sin x\right)}{x}$

$= \lim_{x \to 0}2\left(\tan x - \cos x\right) + x\left(\sec^{2}x + \sin x\right)$

$= - 2$