Answer
$$f'\left( x \right) = \sec x\tan x - \sqrt 2 {\sec ^2}x$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \sec x - \sqrt 2 \tan x \cr
& {\text{find the derivative}} \cr
& f'\left( x \right) = \left( {\sec x - \sqrt 2 \tan x} \right)' \cr
& {\text{sum rule}} \cr
& f'\left( x \right) = \left( {\sec x} \right)' - \left( {\sqrt 2 \tan x} \right)' \cr
& f'\left( x \right) = \left( {\sec x} \right)' - \sqrt 2 \left( {\tan x} \right)' \cr
& {\text{use basic formulas }} \cr
& f'\left( x \right) = \sec x\tan x - \sqrt 2 \left( {{{\sec }^2}x} \right) \cr
& f'\left( x \right) = \sec x\tan x - \sqrt 2 {\sec ^2}x \cr} $$