Question

Differential coefficient of sec(tan⁻¹ x) is ______.

Options

• `x/(1 + x^2)`

• `x sqrt(1 + x^2)`

• `1/sqrt(1 + x^2)`

• `x/sqrt(1 + x^2)`

Answer 1

`x/sqrt(1 + x^2)`

Explanation:

y = sec(tan⁻¹ x)

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

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

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

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

`dy/dx =  y × x × 1/sqrt(1 + x^2)`

`dy/dx = y(x/sqrt(1 + x^2))`

`dy/dx = (x/sqrt(1 + x^2))y`

This is the equation of differential equation which have coefficient `x/sqrt(1 + x^2)`.