The derivative of cosec-1 ((1/2x√1-x2)) w.r.t √1-x2 is

Question

The derivative of cosec-1 ((1/2x√1-x2)) w.r.t √1-x2 is (A) (1/√1 - x2) (B) (2/x) (C)  - (2/x) (D)  - (1/ √1 - x2)

Tardigrade Admin · Accepted Answer

Put x = sinθ Let y = cosec-1 ((1/2x √1-x2)) = cosec -1 ((1/2 sin θ cosθ)) = cosec-1 (cosec 2 θ) = 2 θ ∴ (dy/dz) = (dy/dθ/dz/dθ) = (2/- sinθ) = -(2/x) z = √1-x2 = √1- sin2θ = cosθ

Q. The derivative of $cose c^{- 1} (\frac{1}{2 x 1 - x ^{2}}) w . r . t 1 - x^{2}$ is

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

$\frac{2}{x}$

$- \frac{2}{x}$

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

Put $x = sin θ$
Let $y = cose c^{- 1} (\frac{1}{2 x 1 - x ^{2}})$
$= cose c^{- 1} (\frac{1}{2 s i n θ c o s θ}) = cose c^{- 1}$
$(cosec 2 θ) = 2 θ$
$∴ \frac{d y}{d z} = \frac{d y / d θ}{d z / d θ} = \frac{2}{- s i n θ} = - \frac{2}{x}$
$z = 1 - x^{2} = 1 - sin^{2} θ = cos θ$

Q. The derivative of cosec−1(2x1−x2​1​)w.r.t1−x2​ is

1−x2​1​

x2​

−x2​

−1−x2​1​