TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The chain rule for complex derivatives

Statement

Let $f : U \to V$ and $g : V \to C$ , where $U, V \subseteq C$ are open. If $f$ is complex differentiable at $a \in U$ and $g$ is complex differentiable at $f (a)$ , then $g \circ f$ is complex differentiable at $a$ and

$(g \circ f)^{'} (a) = g^{'} (f (a)) f^{'} (a) .$

Facts & Assumptions

Given: The maps, domains, point, and differentiability hypotheses in the Statement.

[L1]

Complex differentiability at a point is equivalent to real total differentiability with total derivative given by multiplication by the complex derivative (Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with $\partial_{\overset{ˉ}{z}} f = 0$ , or with the Cauchy–Riemann equations).

[L2]

If $f$ is totally differentiable at $a$ and $g$ is totally differentiable at $f (a)$ , then $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

Proof

technique · direct

1.1

By [L1], $D f (a)$ is multiplication by $f^{'} (a)$ and $D g (f (a))$ is multiplication by $g^{'} (f (a))$ .

givenL1

2.1

By [L2], $g \circ f$ is real totally differentiable and its derivative is the composite of the maps in step 1.1, namely multiplication by $g^{'} (f (a)) f^{'} (a)$ .

step 1.1L2algebra

3.1

Applying the reverse implication of [L1] gives complex differentiability of $g \circ f$ and the asserted derivative.

step 2.1L1∎

Depends on

Used by

FALSE: existence of partial derivatives satisfying Cauchy–Riemann everywhere on an open set implies holomorphy False statement

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 30 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Lebl, Guide to Cultivating Complex Analysis, Proposition 2.2.2 (standard reference, not scraped)
J. Orloff, MIT 18.04 Topic 2, §2.6.1 (standard reference, not scraped)