Method of DifferentiationmediumPYQ · JEE Main · 4 Apr 2026 · Shift 1 (Morning)Free

Derivative of Inverse Trig Sum at x=√3/2: 1 | JEE Main 2026

JEE Maths question with a full step-by-step solution.

Question
If y=tan⁡−1(3cos⁡x−4sin⁡x4cos⁡x+3sin⁡x)+2tan⁡−1(x1+1−x2)y=\tan^{-1}\left(\dfrac{3\cos x-4\sin x}{4\cos x+3\sin x}\right)+2\tan^{-1}\left(\dfrac{x}{1+\sqrt{1-x^2}}\right), then dydx\dfrac{dy}{dx} at x=32x=\dfrac{\sqrt3}{2} is equal to
A33
B−1-1
C11correct
D22
Solution
Step 1: Divide numerator and denominator of the first fraction by 4cos⁡x4\cos x:
3cos⁡x−4sin⁡x4cos⁡x+3sin⁡x=34−tan⁡x1+34tan⁡x=tan⁡(tan⁡−134−x).\frac{3\cos x-4\sin x}{4\cos x+3\sin x}=\frac{\frac34-\tan x}{1+\frac34\tan x}=\tan\left(\tan^{-1}\tfrac34-x\right).
 So the first term is tan⁡−134−x\tan^{-1}\dfrac34-x, whose derivative is −1-1. Step 2: For the second term, put x=sin⁡θx=\sin\theta:
2tan⁡−1(sin⁡θ1+cos⁡θ)=2tan⁡−1(tan⁡θ2)=θ=sin⁡−1x,2\tan^{-1}\left(\frac{\sin\theta}{1+\cos\theta}\right)=2\tan^{-1}\left(\tan\tfrac{\theta}{2}\right)=\theta=\sin^{-1}x,
 whose derivative is 11−x2\dfrac{1}{\sqrt{1-x^2}}. Step 3: So dydx=−1+11−x2\dfrac{dy}{dx}=-1+\dfrac{1}{\sqrt{1-x^2}}. At x=32x=\dfrac{\sqrt3}{2}, 1−x2=12\sqrt{1-x^2}=\dfrac12:
dydx=−1+2=1.\frac{dy}{dx}=-1+2=1.
 Correct answer: (3)
Still stuck on this question?Ask your doubt on WhatsApp
Similar questions
Method of Differentiation · easy
If f′(x)=3x2+6f'(x) = \sqrt{3x^2+6} and y=f(x3)y = f(x^3), then at x=1x = 1, (dydx)\left(\dfrac{dy}{dx}\right) is equal to.
Method of Differentiation · easy
If yy is a function of xx and log⁡(x+y)=2xy\log(x+y) = 2xy, then y′(0)y'(0) is.
Method of Differentiation · easy
ddx[sin⁡x+cos⁡x1+sin⁡2x]\dfrac{d}{dx}\left[\dfrac{\sin x + \cos x}{\sqrt{1 + \sin 2x}}\right], where 0<x<π40 < x < \dfrac{\pi}{4}, is:
Method of Differentiation · easy
If y=1+x2+x41+x+x2y=\dfrac{1+x^{2}+x^{4}}{1+x+x^{2}} and dydx=ax+b\dfrac{dy}{dx}=ax+b, then the values of aa and bb are

Solve more, learn faster

Sign up free to solve more JEE Maths questions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.

Sign up freeExplore doMath