નિશ્વાયક — ગણિત ધોરણ 12 સાયન્સ — Question

$f(x)=\begin{vmatrix}\mathbf{\sec x} & \mathbf{x} & \mathbf{1} \\2 \sin x & x^2 & 2x \\ tan x& x & 1\end{vmatrix}= \begin{vmatrix}\mathbf{\sec x-\tan x} & \mathbf{0} & \mathbf{0} \\ 2 \sin x & x^2 & 2x \\ \tan x & x & 1\end{vmatrix} \ \ \ \ \ \ \ \ \ \ \ R_{31}(-1)$

$\therefore f(x)=-x^2 (\sec x-\tan x)$

$\therefore f(x)=-2x (\sec x-\tan x)-x^2 (\sec x \tan x - sec^2x)$

$\therefore \frac{f'(x)}{x}=-2(\sec x-\tan x)-x (\sec x \tan x - \sec^2x)$

$\lim_{x \rightarrow 0}\frac{f'(x)}{x}=\lim_{x \rightarrow 0}[-2(\sec x-\tan x)-x(\sec x \tan x-\sec^2x)]$

$=-2(1-0)-0(0-1)=-2$