If x= , y= ^3 , then d^2 y d x^2 at = 6 is

Step-by-step solution

To compute the second derivative d² y d x² for the parametric equations x = and y = ³ , begin by finding the first derivative. Using the chain rule, d y d x = d y/d d x/d = 3 ² = 3 ² , valid for 0 . For the second derivative, apply d² y d x² = d d ( d y d x ) d d x . Differentiating gives d d (3 ² ) = 6 and d d x = 1 , so d² y d x² = 6 . At = 6 , this evaluates to 6 ( 6 ) = 6 1 2 = 3 . The second derivative at this point is 3 .