Let f(x)=2 sin -1 √1-x+ sin -1(2 √x(1-x)) and g(x)= cos -1((1-x2/1+x2)) then derivative of f(x) w.r.t. g(x) at x=(1/4) is

Question

Let f(x)=2 sin -1 √1-x+ sin -1(2 √x(1-x)) and g(x)= cos -1((1-x2/1+x2)) then derivative of f(x) w.r.t. g(x) at x=(1/4) is (A) (17/64) (B) (17/32) (C) 0 (D) None

Tardigrade Admin · Accepted Answer

Put x= cos 2 θ f ( x )=2 θ+ sin -1( sin 2 θ)=2 θ+π-2 θ=π ∴ f prime((1/4))=0 Hence, derivative of f(x) w.r.t. g(x) at x=(1/4) is 0 .

Q. Let $f (x) = 2 sin^{- 1} 1 - x + sin^{- 1} (2 x (1 - x))$ and $g (x) = cos^{- 1} (\frac{1 - x ^{2}}{1 + x ^{2}})$ then derivative of $f (x)$ w.r.t. $g (x)$ at $x = \frac{1}{4}$ is

$\frac{17}{64}$

$\frac{17}{32}$

0

None

Put $x = cos^{2} θ$
$f (x) = 2 θ + sin^{- 1} (sin 2 θ) = 2 θ + π - 2 θ = π$
$∴ f^{'} (\frac{1}{4}) = 0$
Hence, derivative of $f (x)$ w.r.t. $g (x)$ at $x = \frac{1}{4}$ is 0 .

Q. Let f(x)=2sin−11−x​+sin−1(2x(1−x)​) and g(x)=cos−1(1+x21−x2​) then derivative of f(x) w.r.t. g(x) at x=41​ is

6417​

3217​

0

None

Put x=cos2θ f(x)=2θ+sin−1(sin2θ)=2θ+π−2θ=π ∴f′(41​)=0 Hence, derivative of f(x) w.r.t. g(x) at x=41​ is 0 .

Q. Let $f (x) = 2 sin^{- 1} 1 - x + sin^{- 1} (2 x (1 - x))$ and $g (x) = cos^{- 1} (\frac{1 - x ^{2}}{1 + x ^{2}})$ then derivative of $f (x)$ w.r.t. $g (x)$ at $x = \frac{1}{4}$ is

$\frac{17}{64}$

$\frac{17}{32}$

Put $x = cos^{2} θ$
$f (x) = 2 θ + sin^{- 1} (sin 2 θ) = 2 θ + π - 2 θ = π$
$∴ f^{'} (\frac{1}{4}) = 0$
Hence, derivative of $f (x)$ w.r.t. $g (x)$ at $x = \frac{1}{4}$ is 0 .