Determine if the following functions satisfy local or uniform Lipschitz condition.

1). te^y

My work:
I found d/dy (te^y) = te^y, and this is not bounded above for any value of y, so this made me conclude that it has locally Lipschitz condition since the Lipschitz constant here changes as the reagion changes? Am I right? I used the equation
| f(t,y_1) - f(t, y_2) | = d/dy(f(t,y)) + | y_1 - y_2|
If my work is incorrect, provide the correct answer and approach.

2). y t^2/ (1 + y^2)

I used the same approach here and d/dy (f ) = t^2 - 2y + y^2/ ( 1 + y^2)^2, which is clearly could be bounded above by a constant but this constant changes as the reagion changes so it is local lipschitz. Please check my answer, and if it is incorrect provide the right answer.