lemma A.73 Differentiation under the integral sign

open in the book · appendices/A-long-proofs.tex:4635 · p. 2828

Rests on

depends_on definition 7.97 Partial derivative; gradient ¶
- depends_on definition 7.26 Derivative of a function at a point ¶
  - depends_on definition 7.16 Limit ¶
    - depends_on definition 7.2 Absolute value ¶
    - depends_on definition 7.9 Real function ¶
      - depends_on definition 3.43 Map ¶
  - depends_on definition 7.9 Real function ¶ ↺
- depends_on definition 5.15 Basis ¶
  - depends_on definition 5.12 Subspace generated by a set of vectors ¶
    - depends_on definition 5.5 Linear combination ¶
      - depends_on definition 4.33 Vector space ¶
    - depends_on definition 5.7 Vector subspace ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-3 ¶
  - depends_on definition 5.14 Linear independence ¶
    - depends_on definition 5.5 Linear combination ¶ ↺
depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
- depends_on proposition 7.22 Sequential characterization ¶
  - depends_on definition 7.20 Continuity at a point ¶
    - depends_on definition 7.16 Limit ¶ ↺
    - depends_on equation 7.7 eq:ana-limit-left ¶
    - depends_on equation 7.5 eq:ana-limit-right ¶
  - depends_on definition 7.4 Convergence ¶
    - depends_on definition 7.2 Absolute value ¶ ↺
  - proves proof ch:05-real-analysis@proof-7 ¶
- depends_on theorem 7.7 Bolzano–Weierstrass ¶
  - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
    - depends_on definition A.36 Cut ¶
    - depends_on theorem A.40 Least-upper-bound property ¶
      - depends_on definition A.36 Cut ¶ ↺
      - depends_on definition A.38 Order on $\R$ ¶
      - proves proof app:A-long-proofs@proof-29 ¶
    - proves proof app:A-long-proofs@proof-32 ¶
  - depends_on corollary A.46 Monotone convergence ¶
    - depends_on theorem A.40 Least-upper-bound property ¶ ↺
    - proves proof app:A-long-proofs@proof-33 ¶
  - proves proof ch:05-real-analysis@proof-4 ¶
- proves proof ch:05-real-analysis@proof-10 ¶
depends_on theorem 7.35 Mean value theorem ¶
- depends_on proposition 7.29 Linearity ¶
  - depends_on definition 7.11 Sum of functions ¶
    - depends_on definition 7.9 Real function ¶ ↺
  - depends_on equation 7.14 eq:ana-derivh ¶
  - depends_on proposition 7.6 Algebra of limits ¶
    - depends_on definition 7.4 Convergence ¶ ↺
    - depends_on proposition 7.3 Triangle inequality ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
      - proves proof ch:05-real-analysis@proof-1 ¶
    - proves proof ch:05-real-analysis@proof-3 ¶
  - proves proof ch:05-real-analysis@proof-13 ¶
- depends_on theorem 7.34 Rolle ¶
  - depends_on lemma 7.33 Fermat: interior extremum ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on equation 7.7 eq:ana-limit-left ¶ ↺
      - depends_on equation 7.5 eq:ana-limit-right ¶ ↺
      - proves proof ch:05-real-analysis@proof-6 ¶
    - proves proof ch:05-real-analysis@proof-17 ¶
  - depends_on theorem 7.24 Extreme value theorem ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶
    - depends_on proposition 7.22 Sequential characterization ¶ ↺
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
    - proves proof ch:05-real-analysis@proof-9 ¶
  - proves proof ch:05-real-analysis@proof-18 ¶
- proves proof ch:05-real-analysis@proof-19 ¶
proves proof app:A-long-proofs@proof-53 ¶

Supports

depends_on lemma A.77 Poincaré lemma, converse form ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Partial derivative; gradient	declared	appendices/A-long-proofs.tex:4644
`depends_on`	→	Heine–Cantor: uniform continuity	declared	appendices/A-long-proofs.tex:4644
`depends_on`	→	Mean value theorem	declared	appendices/A-long-proofs.tex:4644
`depends_on`	←	Poincaré lemma, converse form	declared	appendices/A-long-proofs.tex:4947
`proves`	←	app:A-long-proofs@proof-53	declared	appendices/A-long-proofs.tex:4647