In their original paper Green and Griffiths remark that there is a differentiation operation on their jet bundles:
$$ (-)' : \mathcal{J}_{k,m} \to \mathcal{J}_{k+1,m+1} $$
Which they define on p.47 via
$$ \phi'(j_{k+1}(f))(z) = \frac{d}{dz} \phi(j_k(f))(z) $$
for a section $\phi$ of $\mathcal{J}_{k,m}$ and a holomorphic arc $f : \Delta \to X$ where $j_k$ is jet prologation. Why is this well-defined?
Moreover, following Vojta's notes I am thinking of the Green-Griffiths jet bundles as graded parts of the sheaf of algebras $\mathrm{HS}_X$ which represents Hasse-Schmidt derivations. For a ring $A$ this has an explicit presentation,
$$ \mathrm{HS}_A = A[\mathrm{d}_i(a)]_{i > 0, a \in A} / \left( \substack{ \mathrm{d}_i(a + b) - \mathrm{d}_i(a) - \mathrm{d}_i(b) \\ \mathrm{d}_i(ab) - \sum_{p + q = i} \mathrm{d}_p(a) \mathrm{d}_q(b) }\right) $$
Then $\mathcal{J}_{k,m} \subset \mathrm{HS}_X$ is the subsheaf of terms with only $\mathrm{d}_i$ for $i \le k$ and total degree $m$ where $\mathrm{d}_i$ is given degree $i$. For example, differentiation should be a nonlinear map,
$$ \Omega^1_X = \mathcal{J}_{1,1} \to \mathcal{J}_{2,2} $$
which according to Green-Griffiths is supposed to be defined via,
$$ a \mathrm{d} x \mapsto (\mathrm{d}_1 a)(\mathrm{d}_1 x) + a \mathrm{d}_2 x $$
However, this is manifestly not well-defined. Indeed,
$$ \mathrm{d}(xy) \mapsto \mathrm{d}_2 (xy) = (\mathrm{d}_1 x)(\mathrm{d}_1 y) + x \mathrm{d}_2 y + y \mathrm{d}_2 x $$
but also,
$$ \mathrm{d}(xy) = x \mathrm{d} y + y \mathrm{d} x \mapsto 2 (\mathrm{d}_1 x)(\mathrm{d}_1 y) + y \mathrm{d}_2 x + x \mathrm{d}_x y $$
which are not the same? What is going on?
