I'm in doubt about the notation $$ \frac{\partial f(x,b)}{\partial b}(x_0, x_0) = \frac{d}{dx} g(x) (x_0), $$ which can be read as: "Partial derivative of $f(x,b)$ by $b$ in point $(x,b) = (x_0, x_0)$ equals to ordinary derivative of $g(x)$ in point $x_0$. It is assumed that $f,g$ are polynomial functions.
Is it legal to use above approach in terms of Differential equations ?
On
Any notation is legal if it's defined well enough, but I think this might look a bit confusing at first read which is counterproductive to your goals as a writer. You might consider using the subscript notation for partial derivatives $$f_b = \frac{\partial f}{\partial b}$$ for $f = f(x,b)$, and the capital D notation for ordinary derivative $$\text Dg = \frac{\text dg}{\text dx}$$ for $g = g(x)$. This gives the somewhat more readable and concise statement
$$f_b(x_0,x_0) = \text Dg(x_0)$$
The popular PDE text by Lawrence Evans uses this kind of notation, so it has a benefit that many readers will be comfortable with it immediately.
It is legal but maybe the best would be $$ \frac{\partial f(x,b)}{\partial b}\Big\vert_{(x,b)=(x_0, x_0)} = \frac{d}{dx} g(x) \Big\vert_{x=x_0}, $$ or define functions $$ h(x,b)=f_b(x,b)=\frac{\partial f(x,b)}{\partial b} $$ and $$ i(x)=g_x(x)= \frac{d}{dx} g(x), $$ and then refer to $h(x_0,x_0)=f_b(x_0,y_0)$ and $i(x_0)=g_x(x_0)$, $$f_b(x_0,y_0)=g_x(x_0)$$ or $$h(x_0,y_0)=i(x_0)$$