Suppose a set of $n-1$ are given in 2D space, $x_1, x_2, \dots, x_{n-1}$, and an additional point $x_n$ is to be assigned a 2D coordinate such that the prescribed Euclidean distances $d_1, d_2, \dots, d_{n-1}$ of this points to points $x_1, x_2, \dots, x_{n-1}$ are matched as best as possible in least-squares sense.
In other words, a function to be optimized is: $$f(x_n)=\sum_{i=1}^{n-1} (d_i-\|x_i-x_n\|)^2,$$ is minimized. An approach to achieve the minimimum is to adapt some form of iterative minimizers (at least in the literature I have on disposal). However, an approach to minimize the function is to find its derivative wrt $x_n$, set it to zero,$$\frac{\partial}{\partial x_n}f(x_n)=0,$$ and solve for $x_n$. Why is such approach not applicable here? Is is due to hardness of finding a derivative (or impossibility to find one), or due to computational effort?