Parametric equations of tangent line.

What's the parametric equations for the tangent line {x=e^t, y=te^t, z=t^e^t^2} at (1,0,0)

Is it x=1+t, y=(e^1)*t?

Just verifying.

dr/dt =( e^t, e^t+tet, d/dt t^(e^t^2))

z=t^(e^t^2)
Ln z =(e^t^2) /d/dt
(1/z) dz/dt = 2t(e^t^2)
dz/dt=2t t^(e^2t^2)
at x,y,z) = (1,0,0) , taking x=1 , t=0

thus , dr/dt = (1,1,0)
The line ,
(x,y,z) - (1,0,0) = L(1,1,0)
x=1+L
y=L
z=0