Question

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

\\( \frac { d y } { d x } = \\)

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, \(u=\sec x\), \(u^\prime=\sec x\tan x\); \(v = 1+\sec x\), \(v^\prime=\sec x\tan x\).

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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:

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