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 · medium
If y=(sin⁡−1x)2+(cos⁡−1x)2y = (\sin^{-1}x)^2 + (\cos^{-1}x)^2, then (1−x2)d2ydx2−xdydx(1 - x^2)\dfrac{d^2 y}{dx^2} - x\dfrac{dy}{dx} is:
Method of Differentiation · medium
Let f(x)f(x) and g(x)g(x) be twice differentiable with f′′(x)=g′′(x)f''(x)=g''(x) for all x∈Rx\in\mathbb{R}, f′(1)=2g′(1)=4f'(1)=2g'(1)=4, and g(2)=3f(2)=9g(2)=3f(2)=9. Then f(25)−g(25)f(25)-g(25) equals:
Method of Differentiation · medium
Let ff be a real polynomial of degree nn such that f(x)=f′(x) f′′(x)f(x)=f'(x)\,f''(x) for all x∈Rx\in\mathbb{R}. If f(0)=0f(0)=0, then 36(f′(2)+f′′(2)+∫02f(x) dx)36\left(f'(2)+f''(2)+\displaystyle\int_0^{2}f(x)\,dx\right) is equal to
Method of Differentiation · medium
If f(x)=(ax+b)sin⁡x+(cx+d)cos⁡xf(x) = (ax+b)\sin x+(cx+d)\cos x, then the values of a,b,c,da, b, c, d such that f′(x)=xcos⁡xf'(x) = x\cos x for all xx, then (a+b+c+d)(a+b+c+d) is.

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