Remark 88.5.1 (Linear approximation). Let $A$ be a ring and $I \subset A$ be a finitely generated ideal. Let $C$ be an $I$-adically complete $A$-algebra. Let $\psi : A[x_1, \ldots , x_ r]^\wedge \to C$ be a continuous $A$-algebra map. Suppose given $\delta _ i \in C$, $i = 1, \ldots , r$. Then we can consider
\[ \psi ' : A[x_1, \ldots , x_ r]^\wedge \to C,\quad x_ i \longmapsto \psi (x_ i) + \delta _ i \]
see Formal Spaces, Remark 87.28.1. Then we have
\[ \psi '(g) = \psi (g) + \sum \psi (\partial g/\partial x_ i)\delta _ i + \xi \]
with error term $\xi \in (\delta _ i\delta _ j)$. This follows by writing $g$ as a power series and working term by term. Convergence is automatic as the coefficients of $g$ tend to zero. Details omitted.
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