how to calculate $z'(y) $ near $ y = 0$. of the curve $g(t) = (e^t + t,~t^2+3\sin(t)~,t^4+t+1)$

Question

$ Let ~g(t) = (e^t + t,~t^2+3\sin(t)~,t^4+t+1)$

such that $g(0) = (1,0,1)$

calculate$~~z'(y) $ near $ y = 0$.

how do i solve this using implicit function theorem ? it was an exam question , i am not sure if i solved it right.

my trial :

$ \frac{dz}{dy} = \frac{\frac{dz}{dt}}{\frac{dy}{dt}} = \frac{4t^3+1}{2t+3\cos(t)} = \frac{1}{3} ( because~t = 0)$

edit : any conformation of the soultion is it right / wrong ?