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

Step-by-step solution

Let u = ⁻¹ ( 1 + x^2 - 1 x ) and v = ⁻¹ ( x 1 - x^2 ) . Substituting x = in u : u = ⁻¹ ( - 1 ) = ⁻¹ ( 1 - ) u = ⁻¹ ( 2 ^2( /2) 2 ( /2) ( /2) ) = ⁻¹ ( 2 ) = 2 = 1 2 ⁻¹ x Differentiating u with respect to x : du dx = 1 2(1 + x^2) Substituting x = in v : v = ⁻¹ ( ) = ⁻¹( ) = = ⁻¹ x Differentiating v with respect to x : dv dx = 1 1 - x^2 The derivative of u with respect to v is: du dv = du dx dv dx = 1 2(1 + x^2) 1 1 - x^2 = 1 - x^2 2(1 + x^2) Substituting x = 1 2 : du dv = 1 - ( 1 2 )^2 2 (1 + ( 1 2 )^2 ) = 3 4 2 ( 5