If a (4+x^2 )=x and y -x^3=a^2 then dy d x at x=1 is ...

Step-by-step solution

From a(4+x^2) = x , we obtain a = x 4+x^2 . Substituting into y - x^3 = a^2 yields y = x^3 + x^2 (4+x^2)^2 . Differentiating gives dy dx = 3x^2 + d dx ( x^2 (4+x^2)^2 ) . Using the quotient rule: d dx ( x^2 (4+x^2)^2 ) = 2x(4+x^2)^2 - x^2 4x(4+x^2) (4+x^2)^4 = 2x(4+x^2)[(4+x^2) - 2x^2] (4+x^2)^4 = 2x(4-x^2) (4+x^2)^3 . Therefore, dy dx = 3x^2 + 2x(4-x^2) (4+x^2)^3 . Evaluating at x=1 : . dy dx |_ x=1 = 3 + 2(4-1) (4+1)^3 = 3 + 6 125 = 381 125 . Final answer: 381 125