1.

Find \(\frac{d^2y}{dx^2}\), if y=tan^2⁡x+3 tan⁡x.(a) sec^2⁡⁡x tan⁡x (2 tan⁡x+sec⁡x+3)(b) 2 sec^2⁡⁡x tan⁡x (2 tan⁡x-sec⁡x+3)(c) 2 sec^2⁡⁡x tan⁡x (2 tan⁡x+sec⁡x+3)(d) 2 sec^2⁡⁡x tan⁡x (2 tan⁡x+sec⁡x-3)This question was posed to me in an interview.The above asked question is from Second Order Derivatives topic in section Continuity and Differentiability of Mathematics – Class 12

Answer»

The correct answer is (c) 2 sec^2⁡⁡x tan⁡x (2 tan⁡x+sec⁡x+3)

EXPLANATION: Given that, y=tan^2⁡⁡x+3 tan⁡x

\(\FRAC{DY}{DX}\)=2 tan⁡x sec^2⁡⁡x+3 sec^2⁡x=sec^2⁡⁡x (2 tan⁡x+3)

By USING the u.v rule, we get

\(\frac{d^2 y}{dx^2}=\frac{d}{dx}\) (sec^2⁡⁡x).(2 tan⁡x+3)+\(\frac{d}{dx}\) (2 tan⁡x+3).sec^2⁡⁡x

\(\frac{d^2 y}{dx^2}\)=2 sec^2⁡⁡x tan⁡x (2 tan⁡x+3)+sec^2⁡⁡x (2 sec⁡x tanx)

=2 sec^2⁡x tan⁡x (2 tan⁡x+sec⁡x+3).


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