The derivative of ⁻¹ ( 1+x^2 -1 x ) with respect to ⁻¹ ( 2 x 1-x^2 1-2 x^2 ) at x=0 , is

Step-by-step solution

Let y= ⁻¹ ( 1+x^2 -1 x ) Putting x= , we get aligned y & = ⁻¹ ( -1 )= ⁻¹ ( 2 ) & = 1 2 ⁻¹ x aligned On differentiating both sides w.r.t. x , we get array r d y d x = 1 2 (1+x^2 ) and z= ⁻¹ ( 2 x 1-x^2 1-2 x^2 ) array Putting x= , we get aligned & z= ⁻¹ ( 2 2 )= ⁻¹( 2 ) & z=2 =2 ⁻¹ x aligned On differentiating both sides w.r.t. x , we get d z d x = 2 1-x^2 Thus, d y d z = d y / d x d z / d x = 1 4 (1+x^2 ) 1-x^2 [ d y d z ]_ x=0 = 1 4