Let $x_0$ and $x_1$ be two distinct nodes. Let $P(x)$ be a polynomial of degree 2 or less such that :
$P_0' (x_0 ) = f' (x_0 )$ , $P (x_1 ) = f (x_1 )$ , $P_0' (x_1 ) = f' (x_1 )$ , and $P (x) = f' (x_0 )Q_0 (x) + f (x_1 )Q_1 (x) + f' (x_1 )Q_2 (x)$
for some polynomials $Q_0 , Q_1 , Q_2$.
Write down the conditions that determine $Q_0 , Q_1 , Q_2$ (e.g. $Q_1 (x_1 ) = 1$) and then determine $Q_0 , Q_1 , Q_2$ .
My initial approach was to find interpolation of $P'(x)$ from first and third equation then get $P(x)$ by integrating $P'(x)$ determine the constant value from the second equation. But I am not sure if this is the correct approach.