The second derivative of a ^3 t w.r.t a ^3 t at t= 4 is

Step-by-step solution

aligned & d ^2 (a ^3 t ) (a ^3 t )^2 = d ( d (a ^3 t ) d (a ^3 t ) ) d (a ^3 t ) = d d (a ^3 t ) / d t ~d (a ^3 t / d t ) d (a ^3 t ) & = d 3 a ^2 t t -a 3 ^2 t t d (a ^3 t ) = d - t d (a ^3 t ) & = d (- t) d t d (a ^3 t ) d t = - ^2 t -3 a ^2 t t = 1 3 a ^4 t t & at, t= 4 ; 1 3 a ^4 4 4 = 1 3 a ( 1 2 )^4 1 2 = 4 2 3 a aligned