I'm trying to show the existence and uniqueness of an interpolating polynomial $p$ of degree less than or equal to 3 that interpolates a differentiable function $f$ ( i.e. $f(x\_i) = p(x\_i)$ for $i = 0, 1, 2 $ ) and it is also given that that $f'(x\_1) = p'(x\_1)$. 

The way I'm approaching the problem is trying a polynomial of the type:
$ p(x) = a\_0 +a\_1x + a\_2x^2+a\_3x^3 $ and solving for $a\_0, a\_1, a\_2, a\_3$. In this way, I have to show that the determinant of the matrix:

$ A = \begin{bmatrix}1& x\_0 & x\_0^2 &x\_0^3\\1& x\_1 & x\_1^2 &x\_1^3\\1& x\_2 & x\_2^2 &x\_2^3 \\ 0 &1& 2x\_1 & 3x\_1^2 \end{bmatrix} $

is non-zero (for the system to be uniquely determined). I have calculated the value of the determinant using matlab, but the expression was pretty messy and it is not obvious to me that it must be non-zero (for different $x\_0, x\_1, x\_2$). 

Extra: How one can generalize that to a degree $n$ polynomial (i.e. to prove that given $n+1$ independent "conditions" there exists a unique polynomial that satisfies all of them).