If y=t^2+t^3 and x=t-t^4 then d^2 y d x^2 at t=1 is

Step-by-step solution

y=t^2+t^3 aligned & x=t-t^4 d y d t =2 t+3 t^2 & d y d t =2+6 t and d x d t =1-4 t^3 & d^2 x d t^2 =-12 t^2 aligned So, d y d x = 2 t+3 t^2 1-4 t^3 Differentiate w.r.t. x aligned & d^2 y d x^2 = (1-4 t^3 )(2+6 t)- (2 t+3 t^2 ) (-12 t^2 ) (1-4 t^3 )^2 d t d x & d^2 y d x^2 = (1-4 t^3 )(2+6 t)+12 t^2 (2 t+3 t^2 ) (1-4 t^3 )^3 & . d^2 y d x^2 |_ t=1 = (1-4)(2+6)+12(2+3) (1-4)^3 & = (-3)(8)+12(5) (-3)^3 = 36 -27 = -4 3 aligned