This is a derivative problem of multivariate implicit function.

Find dy/dt first, and then find the partial derivatives of t on both sides of the following formula to get e y+te y (dy/dt)+dy/dt = 0.

So substitute t=0 into te y+y+ 1 = 0, and e y+te y (dy/dt)+dy/dt = 0.

You can get dy/dt =-e (- 1) y =- 1.

dx/dt=2t- 1=- 1 x=0

To sum up, the slope of the tangent is (dy/dt)/(dx/dt) = e (- 1).

y+ 1=xe^(- 1)