Answer
$${\mathbf{H}(x, y,z)=\arcsin(yz ) \ \mathbf{i}+ \frac{x z}{\sqrt{(1-( yz)^2}} \mathbf{j} + \frac{xy}{\sqrt{(1-( yz)^2}}\mathbf{k} } $$
Work Step by Step
Given
$$h(x, y,z)=x\ \arcsin (yz)$$
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}= \arcsin (yz) } \\
{h_{y}(x, y,z)=\frac{\partial h(x,y,z)}{\partial y}=x \ \frac{z}{\sqrt{(1-( yz)^2}} = \frac{xz}{\sqrt{(1-( yz)^2}} } \\
{h_{z}(x, y,z)=\frac{\partial h(x,y,z)}{\partial z}= x\ \frac{y}{\sqrt{(1-( yz)^2}} =\frac{xy}{\sqrt{(1-( yz)^2}} } \\ \end{array}
So, we get
${\mathbf{H}(x, y,z)=\arcsin(yz ) \ \mathbf{i}+ \frac{x z}{\sqrt{(1-( yz)^2}} \mathbf{j} + \frac{xy}{\sqrt{(1-( yz)^2}}\mathbf{k} } $
