I do understand Horner's method and I understand how it can be used to efficiently evaluate a polynomial
$$p_n(x) = a_nx^n+...,a_1x+a_0 $$
at $x = x_0$ using only $n$ multiplications. This results in a polynomial $q_{n-1}$ of degree $n-1$ (whose coefficients depend on $x_0$) such that
$$ p_n(x) = (x-x_0)q_{n-1}(x) + p_n(x_0).$$
One can then observe, that
$$ p_n^{\prime}(x_0) = q_{n-1}(x_0) $$
and again apply Horner's method, this time to evaluate $q_{n-1}(x_0)$, in order to find $p_n^{\prime}(x_0)$.
This is frequently done when applying Horner's method to make Newton's root finding algorithm more efficient.
I do not see however, why this way of computing $p_n^{\prime}(x_0)$ via $q_n$ is in any way better than just computing the general derivative function $p_n^{\prime}(x)$ and then use Horner's method to evaluate $p_n^{\prime}(x_0)$.
Why do we make a detour and use the polynomial $q_{n-1}$ ( which results from our application of Horner's method to evaluate $p_n(x_0)$) to compute $p_n^{\prime}(x_0)$ instead of computing it "directly", by determining $p_n^{\prime}(x)$ and then using Horner's method to evaluate $p_n^{\prime}(x_0)$?