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

Step-by-step solution

Given y = ⁻¹ ( 1+x 2 ) + x^x , compute dy dx at x=1 . Let u = ⁻¹ ( 1+x 2 ) and v = x^x . Since ⁻¹( ) = 2 - for [0, ] , with = 1+x 2 , we have u = 2 - 1+x 2 . Differentiate u : du dx = - 1 2 1+x 2 1 2 = - 1 4 1+x 2 . At x=1 , du dx = - 1 4 1 = - 1 4 . For v = x^x , use logarithmic differentiation. Taking v = x x , then 1 v dv dx = x + 1 , so dv dx = x^x ( x + 1) . At x=1 , dv dx = 1^1 ( 1 + 1) = 1 . Combine derivatives: dy dx = du dx + dv dx = - 1 4 + 1 = 3 4 . The final answer is 3 4 .