I have the following equation that predicts a certain parameter $c_t$: $$c_t = f(x,y_1,y_2,y_3...)$$ The function $f$ is known, continuous, real, differentiable to all arguments and the solution to a linear set of differential equations. I want to know $x$, and to do so I measure $c_t$ with known parameters $y_i$ and their respective uncertainties $σ_{c_t}$, $σ_{y_i}$.
If the uncertainty of $x$ was not an issue, I could simply fit $x$ so that $f(..)$ is equal to $c_t$. Then I would use the partial derivative approach to find $σ_{x}$, but to my knowledge that is not applicable since I cannot rewrite the equation to return $x$ as in $x = g(c_t,y_1,y_2,y_3...)$
How can I calculate the expected value and uncertainty of $x$ in this case?
Although I realize that this is not a general solution to this problem, it was possible to approximate $c_t$ as function of $x$ using a first order Taylor expansion in this scenario:
$c_t(x) = f(\bar{x},y_1...) + \frac{\partial f(\bar{x},y_1...)}{\partial x} (x - \bar{x})$
This can be written to return $x$:
$x = (c_t(x) - f(\bar{x},y_1...))/\frac{\partial f(\bar{x},y_1...)}{\partial x}+ \bar{x}$
The uncertainty in $x$ follows directly from applying the partial derivative approach to $x$