"Consider the following non-linear regression model: we have $n$ observations and $k - 1$ explanatory variables, and for each $1 \leq i \leq n$, we have $$ y_i = f(x_i, \beta) = \beta_1 + e^{(x_i)' \cdot \beta^*} $$ where $x_i = (x_{i2}, \ldots, x_{ik})$ and $\beta^* = (\beta_2, \ldots, \beta_k)$.
Give the first order conditions for the NLS estimator $b$ and approximate the solution by performing one step of the Newton-Rhapson method with starting value $b = 0$."
First, I wrote down the function we have to minimize: $$ S(b) = \sum_{i = 1}^n (y_i - b_1 - e^{(x_i)'_2 b^*})^2 $$ Then, the derivative with respect to $b_1$ is simply $$ \sum_{i = 1}^n - 2(y_i - b_1 - e^{(x_i)' b_2}) $$ and the derivative with respect to $b_j$ (for $2 \leq j \leq k$) is $$ \sum_{i = 1}^n - 2(y_i - b_1 - e^{(x_i)' b_2}) \cdot -e^{(x_i)'b_2} \cdot x_{ij} $$ Now the thing is, in order to perform Newton-Rhapson, you have to compute the Hessian matrix and evaluate it at zero. However, this is very cumbersome and it just becomes a very big mess: the element at row $j$, column $r$ is $$ 4 \sum_{i = 1}^n x_{ij} \cdot (e^{(x_i)' b_2})^2 \cdot x_{ir} $$ for $j > 1, r > 1$ (slightly different formula's apply when $j = 1$ or $r = 1$, or both).
I'm sure there has to be an easier way, but I don't know it yet. Any help is greatly appreciated.