If f(x) = ⁻¹ ( 2x 1 + x^2 ) , then f' ( 1 2 ) =

Step-by-step solution

Given f(x) = ⁻¹ ( 2x 1 + x^2 ) Let x = , then = ⁻¹x . Substituting x = in the given function: f(x) = ⁻¹ ( 2 1 + ^2 ) f(x) = ⁻¹( 2 ) For x = 1 2 , = ⁻¹ ( 1 2 ) , which means 2 (- 2 , 2 ) . Therefore, ⁻¹( 2 ) = 2 . f(x) = 2 ⁻¹x Differentiating with respect to x : f'(x) = 2 1 + x^2 Substituting x = 1 2 : f' ( 1 2 ) = 2 1 + ( 1 2 )^2 = 2 1 + 1 4 = 2 5 4 = 8 5 Answer: 8 5