= Solution
Let $D=\sum_pr_pp\in X^{(d)}$. The <derivative of the Abelian sum map> is most naturally stated using <principal parts>. The <tangent space> of the <symmetric product of a curve> at $D$ is $H^0(\mathcal O_D(D))$, and the <tangent space> of the <Jacobian variety> is $H^0(K_X)^*$. With these identifications,
$$
\boxed{(d u_d)_D(v)(\omega)=\sum_p\operatorname{res}_p(v_p\omega),\qquad \omega\in H^0(K_X).}
$$
Its dual is restriction of <holomorphic one-forms> to the length-$d$ <divisor> scheme. In particular,
$$
\boxed{\ker(d u_d)_D^*=H^0(K_X(-D)),\qquad
\operatorname{rank}(d u_d)_D=g-h^0(K_X(-D))=d+1-h^0(\mathcal O_X(D)).}
$$
The last equality is the <Riemann-Roch theorem>. These formulas apply to repeated points as well as distinct ones.
Here is a direct proof of the <repeated-point Abel differential formula>. At a multiplicity-$r$ point, choose a <local coordinate> $t$ with $t(p)=0$. A nearby local <divisor> is described by the roots $t_1,\ldots,t_r$, or, without making an ordering choice, by their elementary symmetric functions $e_1,\ldots,e_r$:
$$
\prod_{i=1}^r(t-t_i)=t^r-e_1t^{r-1}+e_2t^{r-2}-\cdots+(-1)^re_r.
$$
These $e_k$ are smooth <local coordinates> on the <symmetric product of a curve>, including at the repeated <divisor>. Under a first-order deformation, minus the logarithmic variation of this defining polynomial is the <principal part>
$$
v_p=\sum_{k=1}^r(-1)^{k-1}\delta e_k\,t^{-k}\pmod{\mathcal O_{X,p}}.
$$
The sign can be checked at $r=1$: moving the point from $0$ to $\epsilon a$ gives $v=a/t$, and the Abel integral changes by $a\omega(p)$.
Write $\omega=(\sum_{j\ge0}c_jt^j)dt$ and its local primitive as $F(t)=\sum_{j\ge0}c_jt^{j+1}/(j+1)$. The local contribution to the <Abelian sum map> is $\sum_i F(t_i)$. <Newton's identities>, linearized at $e_1=\cdots=e_r=0$, give
$$
d\!\left(\sum_i t_i^k\right)=(-1)^{k-1}k\,de_k\quad(1\le k\le r),
\qquad d\!\left(\sum_i t_i^k\right)=0\quad(k>r).
$$
Substituting in the primitive therefore gives
$$
d\!\left(\sum_iF(t_i)\right)=\sum_{k=1}^r(-1)^{k-1}c_{k-1}\,de_k
=\operatorname{res}_p(v_p\omega).
$$
Summing over the support proves the theorem. In particular, a multiplicity-$r$ point tests the first $r$ coefficients of a <holomorphic one-form>, not just its value. The <residue> pairing between <principal parts> of order at most $r$ and differential jets through order $r-1$ is perfect. Thus the annihilator of the derivative is precisely the <holomorphic one-forms> vanishing to order at least $r_p$ at every $p$, namely $H^0(K_X(-D))$.
There is also a <sheaf cohomology> description that explains its <kernel> geometrically. The sequence
$$
0\longrightarrow\mathcal O_X\longrightarrow\mathcal O_X(D)\longrightarrow\mathcal O_D(D)\longrightarrow0
$$
gives a <connecting homomorphism> $\delta:H^0(\mathcal O_D(D))\to H^1(\mathcal O_X)$. Represent a <principal part> by <meromorphic> lifts on coordinate disks; their differences on overlaps give its <Čech cohomology> class. Under <Serre duality>, pairing this class with $\omega$ is the sum of the <residues> of $v_p\omega$, so $\delta$ is the displayed derivative. Exactness shows that its <kernel> is $H^0(\mathcal O_X(D))/\mathbb C$, of dimension $h^0(\mathcal O_X(D))-1$, in agreement with the tangent directions to the <complete linear system of a divisor> in Solution 2.
Two consequences will be useful below. If $d\ge2g-1$, then $K_X(-D)$ has negative degree and no nonzero sections, so $u_d$ is a <submersion> everywhere. Its image is open, and it is also closed because $X^{(d)}$ is <compact>. Since the <Jacobian variety> is connected, $u_d$ is surjective. Thus the points $u(p)$ generate the <Jacobian variety> as a group. Together with <Abel's theorem>, this identifies the degree-zero <Picard group> with the <Jacobian variety>: every <line bundle> has a <meromorphic> section and thus a <divisor> representation, injectivity is <Abel's theorem>, and surjectivity follows from $u_d(D)-d u(p_0)$.
For $g\ge1$, one can choose $g-1$ distinct points imposing independent conditions on $H^0(K_X)$: at each step a surviving nonzero <holomorphic one-form> has only finitely many zeros, so choose a point where it does not vanish. Their sum $D$ has $h^0(K_X(-D))=1$ and hence $h^0(\mathcal O_X(D))=1$. The derivative has <rank> $g-1$ there. Consequently $W_{g-1}=u_{g-1}(X^{(g-1)})$ is an <irreducible> subvariety of dimension $g-1$. This also covers $g=1$, when $X^{(0)}$ and $W_0$ are points.
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