Question

If y = sec (tan⁻¹x), then `dy/dx` at x = 1 is ______.

Options

• `1/2`

• 1

• `1/sqrt(2)`

• `sqrt(2)`

Answer 1

If y = sec (tan⁻¹x), then `dy/dx` at x = 1 is `underlinebb(1/sqrt(2))`.

Explanation:

Differentiate y with respect to x.

`dy/dx = d/dx[sec(tan^-1x)]`

`dy/dx = sec(tan^-1x).tan(tan^-1x) .  d/dx (tan^-1x)`

`dy/dx = sec(tan^-1x) .  x  .  1/(1 + x^2)`

∴ `(dy/dx)_("at"  x  =  1) = sec(tan^-1 1) xx 1 xx (1)/(1 + 1^2)`

= `sec  π/4 xx (1)/(2)`

= `sqrt(2) xx (1)/2`

 = `1/sqrt(2)`