Solution
= Solution
Ignoring the bounded second-order remainder, this is the <linear regression> model
$$
Y_i-g(x_0)=hZ_i\beta+\varepsilon_i,
\qquad \beta=g'(x_0).
$$
The <ordinary least squares> estimate through the known zero intercept is
$$
\widehat\beta
=\frac{\sum_i(hZ_i)(Y_i-g(x_0))}{\sum_i(hZ_i)^2}
=\boxed{\frac1N\sum_{i=1}^N\frac{Z_i(Y_i-g(x_0))}{h}},
$$
because $Z_i^2=1$. This is the <randomized symmetric finite-difference derivative estimator>.