Let $f(u,v)$ be a differentiable function of $u$ and $v$, where $u$ and $v$ are some function of $x$ and $y$. Then let's say $f_u(1,1)=a$ and $f_v(1,1)$= b.
Then let's say I want to find $f_x(x,y)$ at the point $(2,3)$. I will use the chain rule:
$$\frac{\partial f}{\partial x}=\frac{\partial f}{\partial u} \cdot \frac{\partial u}{\partial x} + \frac{\partial f}{\partial v} \cdot \frac{\partial v}{\partial x}$$
My question is, am I able to use the value $f_u(1,1)=a$ in my calculation even though it is not evaluated in the point of interest?