The derivative of ( ⁻¹ [ x 1+ 1-x^2 ] ) with respect to ( ⁻¹ ( 1 2 x^2-1 ) ) is

Step-by-step solution

Let ( aligned y & = ⁻¹ ( x 1+ 1-x^2 ) u & = ⁻¹ ( 1 2 x^2-1 ) aligned ) Let (x= ) ( aligned & y= ⁻¹ ( 1+ ), u= ⁻¹( 2 ) & y= ⁻¹ ( 1 + ),(u=2 ) & y= ⁻¹( - ),(u=2 ) & y= - , u=2 & ^2 y d y d = - ^2 , d u d =2 & d y d = - ^2 ^2 y , d u d =2 & d y d = - ^2 1+( - )^2 , d u d =2 & d y d u = 1 2 [ - ^2 1+ ^2 + ^2 -2 ] & = 1 2 ( - ) 2 ^2 -2 d y d u =- 1 4 aligned )