Answer
${\mathbf{H}(x, y,z)=\left(y \ \ln(x+y) +\frac{xy}{ (x+y)} \right)\mathbf{i}+ \left (x \ \ln(x+y) +\frac{xy}{ (x+y)}\right) \mathbf{j} } $
Work Step by Step
Given
$$h(x, y,z)=xy\ \ln(x+y)$$
Since \begin{align}\mathbf{H}(x,y,z)&=M \mathbf{i}+N\mathbf{j}+P\mathbf{k}\\
&=h_{x}(x, y,z)\mathbf{i}+h_{y}(x, y,z)\mathbf{j}+h_{z}(x, y,z)\mathbf{k}
\end{align}
As, we have
\begin{array}{l}
{h_{x}(x, y,z)=\frac{\partial h(x,y,z)}{\partial x}=y \ \ln(x+y) +\frac{xy}{ (x+y)} } \\
{h_{y}(x, y,z)=\frac{\partial h(x,y,z)}{\partial y}=x \ \ln(x+y) +\frac{xy}{ (x+y)} } \\
{h_{z}(x, y,z)=\frac{\partial h(x,y,z)}{\partial z}=0 } \\ \end{array}
So, we get
${\mathbf{H}(x, y,z)=\left(y \ \ln(x+y) +\frac{xy}{ (x+y)} \right)\mathbf{i}+ \left (x \ \ln(x+y) +\frac{xy}{ (x+y)}\right) \mathbf{j} } $
