Question

Differentiate with respect to x :

If y = sin-1 ( 2x / 1 + x2 ) + sec-1 ( 1 + x2 / 1 - x2 ) , show that dy/dx = 4 / ( 1 + x2 )

Answer 1

Let y = sin {(1-x^2)/(1+x^2)}

dy/dx =cos {(1-x^2)/(1+x^2)}d/dx{(1-x^2)/(1+x^2)(applying chain rule)

= cos{(1-x^2)/(1+x^2)}{(1+x^2)(-2x)-(1-x^2)(2x)}/(1+x^2)^2

= cos{(1-x^2)/(1+x^2)}{-2x-2x^3 - 2x+2x^3}/(1+x^2)^2

= cos{(1-x^2)/(1+x^2)}{-4x/(1+x^2)^2}

** We know ; d/dx (sinx) = cosx and

d/dx(u/v) = (v*u’ -y*v’)/v^2