How to calculate the uncertainty on a function where the coefficients themselves have their own uncertainties?

Solution 1:

Propagation of error formulae assume the uncertainty is in the variables, not the coefficients.

Regardless, if we assume you have no error in $x$ and A and B are uncorrelated random variables then you simply apply the usual algebra of random variables to get the variance of $Y$:

$$Var[A+Bx] = Var[A]+x^2Var[B]$$

In general, for a random variable $Z$ and constant $a$:

$$Var[aZ]=a^2Var[Z]$$

I’d you have a linear combination of uncorrellated random variables you can apply this to each:

$$Var\left[\sum a_iZ_i\right]=\sum a_i^2Var[Z_i]$$

I’m your case, the constants are the values of $x, x^2,…$