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Let y=sec(tan⁻¹x). The value of (dy)/(dx) when x=1 is:

1. (1)/(√(2))
2. (1)/(2)
3. 1
4. √(2)

Correct answer: (1)/(√(2))

Solution

To find ( rac{dy}{dx}) for ( y = ext{sec}( an⁻¹x)), we first express ( y) in terms of ( x) using trigonometric identities. By differentiating and evaluating at ( x = 1, we find that ( rac{dy}{dx} = rac{1}{ ext{sec}( an⁻¹(1)) imes ext{tan}( an⁻¹(1))} = rac{1}{ rac{ ext{sqrt}(2)}{2} imes 1} = rac{1}{ rac{ ext{sqrt}(2)}{2}} = rac{1}{ ext{sqrt}(2)}), confirming option A as correct.

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