= Solution
Let $X$ be a <smooth projective curve> over $\mathbb C$, let $D=\sum_p m_pp$ be an effective <divisor on an algebraic curve> of degree $d$, and identify a translate of the <Jacobian variety> with $\operatorname{Pic}^d(X)$. The precise derivative theorem is
$$
\boxed{T_DX^{(d)}=H^0(X,\mathcal O_D(D)),\qquad d u_d|_D=\delta_D,}
$$
where $\delta_D$ is the <connecting homomorphism> of
$$
0\longrightarrow\mathcal O_X\longrightarrow\mathcal O_X(D)\longrightarrow\mathcal O_D(D)\longrightarrow0.
$$
The target is $T_{u_d(D)}\operatorname{Jac}(X)=H^1(X,\mathcal O_X)=H^0(X,K_X)^*$ under <Serre duality>. Its dual is restriction of <holomorphic differential forms> to the length-$d$ <closed subscheme> $D$:
$$
(d u_d|_D)^*:H^0(K_X)\longrightarrow H^0(K_X|_D).
$$
In particular, this includes derivatives of the local coefficients at repeated points, rather than simply evaluating once for each point of the support.
Here is a local proof of all these identifications. Near a point of multiplicity $m$, take a coordinate $z$ vanishing at that point. Nearby effective <divisors> are represented by monic polynomials of degree $m$. A first-order deformation has equation
$$
z^m-\varepsilon v(z)=0,\qquad v(z)=v_0+v_1z+\cdots+v_{m-1}z^{m-1},\qquad\varepsilon^2=0.
$$
These coefficients are the elementary symmetric coordinates on the <symmetric product of a curve>; they give $m$ independent tangent directions even at a repeated point. Dividing the variation of the equation by the original equation identifies the tangent direction with the <principal part> $v(z)/z^m$. Doing this at each point gives $H^0(\mathcal O_D(D))$, a space of dimension $d$.
To see the derivative of the <line bundle> directly, use the local frame $1/(z^m-\varepsilon v)$ for $\mathcal O_X(D_\varepsilon)$ and the frame $1$ outside the small disks. Relative to the frame $1/z^m$ at $\varepsilon=0$, its transition multiplier changes by
$$
\frac{z^m}{z^m-\varepsilon v}=1+\varepsilon\frac{v}{z^m}.
$$
First-order changes of <line bundle> transition functions have the form $1+\varepsilon c_{ij}$; changing frames adds a <Čech coboundary>. Thus the tangent space to the <Picard group> is $H^1(\mathcal O_X)$, and the derivative is exactly the <Čech cocycle> of these <principal parts>, namely $\delta_D$.
This agrees with differentiating the Abel integrals, including the sign. If $\omega=f(z)\,dz$ and $F'=f$, the sum of $F$ at the roots of $z^m-\varepsilon v$ has derivative
$$
\operatorname{res}_{z=0}\left(\frac{v(z)}{z^m}\omega\right).
$$
For example, expand $F=\sum_{k\ge1}f_{k-1}z^k/k$. The first-order Newton identities give $\partial_\varepsilon\sum z_i^k=k v_{m-k}$ for $1\le k\le m$, and zero for $k>m$, yielding exactly the <residue> above. Equivalently the same formula follows by differentiating the contour integral for the sum over roots. <Serre duality> pairs the <principal part> class with $\omega$ by the sum of these <residues>. Locally this pairing is perfect between the <principal parts> $z^{-1},\ldots,z^{-m}$ and the jets $1,z,\ldots,z^{m-1}$ times $dz$. This proves the asserted dual restriction map.
The <long exact sequence in sheaf cohomology> now gives the kernel and rank without any reduced-support assumption:
$$
\ker(d u_d|_D)=H^0(\mathcal O_X(D))/\mathbb C,\qquad
\boxed{\operatorname{rank}(d u_d|_D)=d+1-h^0(\mathcal O_X(D)).}
$$
By duality,
$$
\boxed{\operatorname{im}(d u_d|_D)=H^0(K_X(-D))^\perp.}
$$
This is the geometric form of the <derivative of the Abelian sum map>. For distinct points, the image is spanned by the evaluation vectors of the <canonical map>; at repeated points, the corresponding osculating directions must also be included. The <Riemann-Roch theorem> confirms the rank as $g-h^0(K_X(-D))$. When $h^0(D)=1$, the <Abelian sum map> is an immersion at $D$. For $d=g-1$ and $h^0(D)=1$, its image is a hyperplane annihilated by the unique nonzero <holomorphic differential form> vanishing along $D$, which will be the tangent hyperplane to the <theta divisor>.
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