Let $f:[a,b] \to \mathbb R$ continous differentiable, each $x_i$ different from each other and let $q_i:x\to \prod_{j\ne i} (x-x_j)^2 $ the square of the not normalized $i$-th Lagrange-polynom.
Show that $$g_i:x\to\frac{q_i(x)}{q_i(x_i)}\Bigg(\Bigg(1-(x-x_i)\frac{q_i'(x_i)}{q_i(x_i)}\Bigg)f(x_i)+(x-x_i)f'(x_i)\Bigg)$$ interpolates $f$ and $f'$ in $x_i$ and that at every other $x_j$ disappearing derivations and values has.
I am absolutly cluess here. Some help is welcome!
Some hints: For $j\neq i$ use that $g_i(x) = (x-x_j)^2 R(x)$ where $R$ is differentiable at $x_j$.
At $x_i$ clearly the value of $g_i(x_i)=f(x_i)$.
For the derivative at $x_i$, it's the derivative of a product. So calculate it and note that there are two terms that cancel nicely to leave you with $f'(x_i)$.