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← JEE Main 2022 · 26 Jul · Shift 2
MathematicsSingle correct+4 / -1Question 8 of 90

JEE Main 2022 · 26 Jul · Shift 2, Mathematics question 8

The value of log⁡e2 ddx(log⁡cos⁡xcosec⁡x)\log_e2\,\dfrac{d}{dx}\left(\log_{\cos x}\operatorname{cosec}x\right) at x=π4x=\dfrac\pi4 is

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Answer and worked solution

Official answer: D

Write g=log⁡cos⁡xcosec⁡x=−ln⁡sin⁡xln⁡cos⁡xg=\log_{\cos x}\operatorname{cosec}x=-\frac{\ln\sin x}{\ln\cos x}.

Differentiate. g′=−cot⁡xln⁡cos⁡x+tan⁡xln⁡sin⁡x(ln⁡cos⁡x)2g'=-\frac{\cot x\ln\cos x+\tan x\ln\sin x}{(\ln\cos x)^2}. At π4\frac\pi4, ln⁡sin⁡=ln⁡cos⁡=−12ln⁡2\ln\sin=\ln\cos=-\frac12\ln2 and tan⁡=cot⁡=1\tan=\cot=1: g′=−2−12ln⁡2=4ln⁡2g'=\frac{-2}{-\frac12\ln2}=\frac4{\ln2}.

Multiplying by ln⁡2\ln2 gives 4. (D)

Topics:Derivatives of standard functionsLogarithms
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