Look at the simple case of

and

. If

then, differentiating,

. Set x= 0 in both equations:

and

. Those are clearly independent equations and have only

as solution.

But I think you are misunderstanding the hint. In this simple example, the determinant of the coefficients is

not 0.

They are saying that if

**any** system of equations has more than one solution, then the determinant of the coefficients must be 0 (if it were not, the matrix of determinants would have an inverse). The point here is to show that the determinant of the coefficients is NOT 0 so there is only the trivial solution.

In general you have

and you want to show that, as long as all of the numbers

are distinct, that is NOT 0.