find \\( \\frac { d y } { d x } \\) for \\( y = \\frac { \\sec x } { 1 + \\sec x } \\).\n\n\\( \\frac { d y…

Explanation:

Step1: Apply quotient rule

The quotient rule states that if (y = \frac{u}{v}), then (y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}). Here, (u=\sec x), (u^\prime=\sec x\tan x); (v = 1+\sec x), (v^\prime=\sec x\tan x). [ \begin{align*} \frac{dy}{dx}&=\frac{\sec x\tan x(1 + \sec x)-\sec x(\sec x\tan x)}{(1 + \sec x)^{2}}\ \end{align*} ]

Step2: Simplify the numerator

Expand the numerator: (\sec x\tan x+\sec^{2}x\tan x-\sec^{2}x\tan x=\sec x\tan x) [ \frac{dy}{dx}=\frac{\sec x\tan x}{(1 + \sec x)^{2}} ]

Answer:

(\frac{\sec x\tan x}{(1 + \sec x)^{2}})