In the paper _[Semi-supervised learning by mixed label propagation][1]_ Wei Tong and Rong Jin define 
  
$S$ as the similarity(adjacency) matrix  
$D = diag(D_1, D_2, ..., D_n)$ where $D_i = \sum_{j=1}^nS_{i,j}$  
  class assignment $\textbf{y} = (y_1, y_2, ..., y_n)$  

Given the class labels of $n_l$ examples, $\widehat{y}_l = (\widehat{y}_1, \widehat{y}_2,...,\widehat{y}_{nl})$, the optimal class assignment **y** is found by minimizing the energy function
 
 [![problem][2]][2]

  where $E(S, \textbf{y}) = \sum_{i, j=1}^n S_{ij}(y_i - y_j)^2 = \textbf{y}^TL
\textbf{y}, L = D - S$. 
How to find $y_u$? I have tried the following.  

$E(S, \textbf{y}) = \begin{bmatrix}
y_l & y_u
\end{bmatrix} 
\begin{bmatrix}
L^{l, l} & L^{l, u}\\
L^{u, l} & L^{u, u}\\
\end{bmatrix} 
\begin{bmatrix}
y_l \\
y_u
\end{bmatrix}
$
  
$ = \begin{bmatrix}
y_l & y_u
\end{bmatrix} 
\begin{bmatrix}
L^{l, l}y_l + L^{l, u}y_u\\
L^{u, l}y_l + L^{u, u}y_u\\
\end{bmatrix} 
$
  
$ = y_l^TL^{l, l}y_l + y_l^TL^{l, u}y_u + y_u^TL^{u, l}y_l + y_u^TL^{u, u}y_u$

when I take the derivative w.r.t $y_u$  I don't understand how to differentiate $y_u^TL^{u, l}y_l$. Following this [method][3], 
  
$(y_u + h)^TL^{u, l}y_l - y_u^TL^{u, l}y_l = h^TL^{u, l}y_l$  

Now, how to divide by $h$. 

Also if this is not the correct way, how to find $y_u$ then?   

  [1]: https://dl.acm.org/doi/10.5555/1619645.1619750
  [2]: https://i.sstatic.net/ivb0M.png
  [3]: https://math.stackexchange.com/a/312083/336994