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JEE Main 2013

JEE Main 2013 Mathematics question on Limits Continuity and Differentiability

If y=sec⁡(tan⁡−1x)y = \sec(\tan^{-1} x), then dydx\dfrac{dy}{dx} at x=1x = 1 is equal to
  1. A.
    12\dfrac{1}{\sqrt{2}}
  2. B.
    12\dfrac{1}{2}
  3. C.
    11
  4. D.
    2\sqrt{2}
Show answer and solution

Correct option: A

Solution

Let t = tan⁻¹x, so y = sec(t). Since tan(t) = x, we get sec(t) = √(1 + x²), so y = √(1 + x²).

dy/dx = x/√(1 + x²).

At x = 1: dy/dx = 1/√2.

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