If $y=\tan ^{-1}\left(\frac{3 \cos x-4 \sin x}{4 \cos x+3 \sin x}\right)+2 \tan ^{-1}\left(\frac{x}{1+\sqrt{1-x^{2}}}\right)$, then $\frac{dy}{dx}$ at $x=\frac{\sqrt{3}}{2}$ is equal to :

JEE Main · 2026 · 04 Apr 2026 · Shift 1 · Inverse Trigonometric Functions

y = tan^{- 1} (\frac{3 c o s x - 4 s i n x}{4 c o s x + 3 s i n x}) + 2 tan^{- 1} (\frac{x}{1 + 1 - x ^{2}})

, then

\frac{d y}{d x}

x = \frac{3}{2}

is equal to :

Answer: (C)

Writing

\frac{3 cos x - 4 sin x}{4 cos x + 3 sin x} = \frac{\frac{3}{4} - tan x}{1 + \frac{3}{4} tan x}

, the first term equals

tan^{- 1} \frac{3}{4} - x

near the point of interest, with derivative

- 1

. For the second term put

x = sin θ

; then

\frac{x}{1 + 1 - x ^{2}} = \frac{sin θ}{1 + cos θ} = tan \frac{θ}{2}

, so

2 tan^{- 1} (tan \frac{θ}{2}) = θ = sin^{- 1} x

, with derivative

\frac{1}{1 - x ^{2}}

. Hence

\frac{d y}{d x} = - 1 + \frac{1}{1 - x ^{2}}

, and at

x = \frac{3}{2}

\frac{d y}{d x} = - 1 + \frac{1}{1 - 3/4} = - 1 + 2 = 1

JEE Main · 2026 · 04 Apr 2026 · Shift 1 · Single correct option

If the domain of the function $f (x) = cos^{- 1} (\frac{2 x - 5}{11 - 3 x}) + sin^{- 1} (2 x^{2} - 3 x + 1)$ is the interval $[α, β]$ , then $α + 2 β$ is equal to :
JEE Main 2026View solution →
The number of solutions of $tan^{- 1} 4 x + tan^{- 1} 6 x = \frac{π}{6}$ , where $- \frac{1}{2 6} < x < \frac{1}{2 6}$ is equal to
JEE Main 2026View solution →
If the domain of the function $f (x) = sin^{- 1} (\frac{1}{x ^{2} - 2 x - 2})$ , is $(- \infty, α] \cup [β, γ] \cup [δ, \infty)$ , then $α + β + γ + δ$ is equal to
JEE Main 2026View solution →
If $k = tan (\frac{π}{4} + \frac{1}{2} cos^{- 1} (\frac{2}{3})) + tan (\frac{1}{2} sin^{- 1} (\frac{2}{3}))$ then the number of solutions of the equation …
JEE Main 2026View solution →
Considering the principal values of inverse trigonometric functions, the value of the expression $tan (2 sin^{- 1} (\frac{2}{13}) - 2 cos^{- 1} (\frac{3}{10}))$ is equal to :
JEE Main 2026View solution →
Let [ • ] denote the greatest integer function. If the domain of the function $f (x) = cos^{- 1} (\frac{4 x + 2 [ x ]}{3})$ is $[α, β]$ , then $12 (α + β)$ is equal to :
JEE Main 2026View solution →