PrepRung
Sign inSign up free

NDA 2020 Maths · Mathematics · Derivatives · medium

If y=ln tan ((π)/(4)+x/2), then dy/dx equals

Single correct option

If y=ln⁡tan⁡(π4+x2)y=\ln\tan\left(\dfrac\pi4+\dfrac x2\right), then dydx\dfrac{dy}{dx} equals
  1. A. csc⁡x\csc x
  2. B. sec⁡x\sec x
  3. C. tan⁡x\tan x
  4. D. sec⁡xtan⁡x\sec x\tan x
Show answer and solution

Answer: (B) sec⁡x\sec x

By the chain rule, dydx=1tan⁡(π/4+x/2)⋅sec⁡2(π4+x2)⋅12=12sin⁡(π/4+x/2)cos⁡(π/4+x/2)=1sin⁡(π/2+x)=1cos⁡x=sec⁡x\dfrac{dy}{dx}=\dfrac{1}{\tan(\pi/4+x/2)}\cdot\sec^2\left(\dfrac\pi4+\dfrac x2\right)\cdot\dfrac12=\dfrac{1}{2\sin(\pi/4+x/2)\cos(\pi/4+x/2)}=\dfrac{1}{\sin(\pi/2+x)}=\dfrac1{\cos x}=\sec x. The distractor sec⁡xtan⁡x\sec x\tan x is the derivative of sec⁡x\sec x, and csc⁡x\csc x results from using sin⁡(π/2+x)=sin⁡x\sin(\pi/2+x)=\sin x by mistake. The identity sin⁡2A=2sin⁡Acos⁡A\sin2A=2\sin A\cos A does the work.
Report a mistake

Original question written in the NDA 2020 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake