If y = ⁻¹ ( 12 x-64 x^3 1-48 x^2 ) , then dy d x =

Step-by-step solution

Given y= ⁻¹ ( 12x-64x^3 1-48x^2 ) , we recognize the expression inside the inverse tangent resembles the triple angle formula for tangent: (3 ) = 3 - ^3 1 - 3 ^2 . Let 4x = , so x = 1 4 . Substituting this into y gives: y = ⁻¹ ( 3 - ^3 1 - 3 ^2 ) . This simplifies to y = ⁻¹( (3 )) = 3 , where = ⁻¹(4x) . Thus, y = 3 ⁻¹(4x) . Differentiating using the chain rule: dy dx = 3 1 1+(4x)^2 4 = 12 1+16x^2 . The result matches option C .