I need to resolve the following exercise
Let $f:A\subset\mathbb{R}^2\rightarrow\mathbb{R}$, $A$ is an open set. Suppose the $f_x(x, y)$ exists for all $(x, y)\in A$. Consider $M(x_0+h,y_0)$ and $N(x_0,y_0)$ two point in $A$. Prove that if line segment with $M$ and $N$ is fully contained in $A$, then there exists an $\overline{x}$ between $x_0+h$ and $x$ so that $$f(x_0+h,y_0)-f(x_0,y_0)=\frac{\partial f}{\partial x}(\overline{x}, y_0)h.$$
I found some demonstrations that use the chain rule but in my book this just comes out 2 chapters later.
In order to resolve that, can I simply define $g(x) :=f(x, y_0)$ as function only $x$ and apply MVT for single variable function? Is it fair?