Question

If y = `cos^-1 sqrt((1 + x^2)/2`, then `dy/dx` = ______.

Options

• `(-1)/(2sqrt(1 - x^4)`

• `1/(2sqrt(1 - x^4)`

• `(-x)/sqrt(1 - x^4)`

• `x/sqrt(1 - x^4)`

Answer 1

If y = `cos^-1 sqrt((1 + x^2)/2`, then `dy/dx` = `underlinebb((-x)/sqrt(1 - x^4))`.

Explanation:

y = `cos^-1 sqrt((1 + x^2)/2`

Put x² = cos 2θ,

∴ θ = `1/2 cos^-1x^2`

Now y = `cos^-1 sqrt((1 + cos2θ)/2`

= `cos^-1 sqrt(cos^2θ)`

∴ y = cos^–1 cos θ

= θ 

= `1/2 cos^-1x^2`

∴ `dy/dx = (-1(2x))/(2sqrt(1 - x^4))`

= `(-x)/sqrt(1 - x^4)`