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

Step-by-step solution

The derivative of f(x) = 1 + ^2(x^2) follows from the chain rule: f'(x) = 1 2 1 + ^2(x^2) d dx [1 + ^2(x^2)] Differentiating the inner function yields d dx [ ^2(x^2)] = 2 (x^2)(- (x^2))(2x) = -4x (x^2) (x^2) Applying the identity 2 = (2 ) gives d dx [ ^2(x^2)] = -2x (2x^2) Thus f'(x) = -x (2x^2) 1 + ^2(x^2) Evaluating at x = 2 : x^2 = 4 and 2x^2 = 2 Numerator: - 2 ( 2 ) = - 2 Denominator: 1 + ^2 ( 4 ) = 1 + ( 1 2 )^2 = 3 2 f' ( 2 ) = - 2 3 2 = - 2 2 3 = - 2 2 3 Rationalizing the denominator gives - 6 6 = - 6 Final