If x = a ^3 t and y = a ^3 t , then the value of d^2y dx^2 at t = 3 is equal to

Step-by-step solution

Given x = a ^3 t and y = a ^3 t . Differentiating with respect to t : dx dt = 3a ^2 t t dy dt = -3a ^2 t t Now, dy dx = dy dt dx dt = -3a ^2 t t 3a ^2 t t = - t Differentiating again with respect to x : d^2y dx^2 = d dt (- t) dt dx d^2y dx^2 = ^2 t 1 3a ^2 t t = 1 3a ^4 t t Substituting t = 3 : ( 3 ) = 3 2 and ( 3 ) = 1 2 d^2y dx^2 = 1 3a ( 3 2 )^4 ( 1 2 ) = 1 3a ( 9 16 ) ( 1 2 ) = 32 27a Answer: 32 27a