Derivative of sin-1 x w.r.t cos-11-x2 is

Let f=sin-1x, g=cos-1 1-x2 Put x=sin θ f1=11-x2 g=cos-1 1-sin2 θ =cos-1 (cos θ) =θ =sin-1 x g1(x) = 11-x2 derivative of f(x) w.r.t g(x) =f1(x)g1(x)=1

Derivative of sin-1 x w.r.t cos-11-x2 is

Let f=sin-1x, g=cos-1 1-x2 Put x=sin θ f1=11-x2 g=cos-1 1-sin2 θ =cos-1 (cos θ) =θ =sin-1 x g1(x) = 11-x2 derivative of f(x) w.r.t g(x) =f1(x)g1(x)=1

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Derivative of sin^-1 x w.r.t cos^-1 $\sqrt{1 - x^{2}}$ is

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Detailed Solution

$Let f = \sin^{- 1} x, g = \cos^{- 1} \sqrt{1 - x^{2}} Put x = \sin θ f^{1} = \frac{1}{\sqrt{1 - x^{2}}} g = \cos^{- 1} \sqrt{1 - \sin^{2} θ} = \cos^{- 1} (\cos θ) = θ = \sin^{- 1} x g^{1} (x) = \frac{1}{\sqrt{1 - x^{2}}} derivative of f (x) w . r . t g (x) = \frac{f^{1} (x)}{g^{1} (x)} = 1$