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Question

Mathematics Question on Continuity and differentiability

Differential coefficient of secx\sqrt{sec\sqrt{x}} is

A

14xsecx sinx\frac{1}{4\sqrt{x}}sec\sqrt{x}\,sin\sqrt{x}

B

14x(secx)3/2⋅sinx\frac{1}{4\sqrt{x}}\left(sec\sqrt{x}\right)^{3/2}\cdot sin\sqrt{x}

C

12xsecxsinx\frac{1}{2}\sqrt{x}sec\sqrt{x}sin\sqrt{x}

D

12x(secx)3/2⋅sinx\frac{1}{2}\sqrt{x}\left(sec\sqrt{x}\right)^{3/2}\cdot sin\sqrt{x}

Answer

14x(secx)3/2⋅sinx\frac{1}{4\sqrt{x}}\left(sec\sqrt{x}\right)^{3/2}\cdot sin\sqrt{x}

Explanation

Solution

Let y=secxy=\sqrt{sec\sqrt{x}} Differentiating ww.rr.tt. xx, we get dydx=12secx⋅secx⋅tanx⋅12x\frac{dy}{dx}=\frac{1}{2\sqrt{sec\sqrt{x}}}\cdot sec\sqrt{x}\cdot tan\sqrt{x}\cdot\frac{1}{2\sqrt{x}} =14x(secx)1/2sinxcosx=\frac{1}{4\sqrt{x}}\left(sec\sqrt{x}\right)^{1/2} \frac{sin\sqrt{x}}{cos\sqrt{x}} =14x(secx)3/2⋅sinx=\frac{1}{4\sqrt{x}}\left(sec\sqrt{x}\right)^{3/2}\cdot sin\sqrt{x}