proposition 7.32 Derivative of the inverse function

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:617 · p. 214

Rests on

depends_on definition 7.14 Inverse function ¶
- depends_on definition 7.13 Composite function ¶
  - depends_on definition 7.9 Real function ¶
    - depends_on definition 3.43 Map ¶
      - depends_on definition 3.24 Quantifiers ¶
      - depends_on definition 3.28 Set ¶
- depends_on definition 3.47 Bijective map ¶
depends_on proposition 7.6 Algebra of limits ¶
- depends_on definition 7.4 Convergence ¶
  - depends_on definition 7.2 Absolute value ¶
- 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 ¶
depends_on theorem 7.23 Intermediate value theorem ¶
- depends_on axiom 7.1 Completeness of $\R$ ¶
- depends_on definition 7.20 Continuity 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 equation 7.7 eq:ana-limit-left ¶
  - depends_on equation 7.5 eq:ana-limit-right ¶
- proves proof ch:05-real-analysis@proof-8 ¶
proves proof ch:05-real-analysis@proof-16 ¶

Supports

depends_on definition 9.140 Amplitude and the Jacobi elliptic functions ¶
- depends_on proposition 9.143 A cubic quadrature between two turning points ¶
- depends_on proposition 9.141 First properties ¶
  - depends_on proposition 9.143 A cubic quadrature between two turning points ¶ ↺
  - depends_on proposition 9.142 The differential equation of the sine amplitude ¶
- depends_on proposition 9.144 The pendulum equation at finite amplitude ¶
depends_on proposition 7.61 The logarithm ¶
- depends_on corollary 7.69 $\ee$ is the unique self-reproducing base ¶
  - depends_on remark 7.70 What the condition says ¶
- depends_on definition 7.62 Real powers ¶
  - depends_on definition 7.90 The golden ratio ¶
    - depends_on proposition 7.91 The golden ratio ¶
      - depends_on proposition 7.92 Continued fraction and the Fibonacci ratios ¶
      - depends_on proposition 125.2 Crystallographic restriction ¶
      - depends_on remark 7.95 Where $\varphi$ is used ¶
      - depends_on remark 7.94 Three numbers, two kinds ¶
  - depends_on proposition 7.63 Laws of real powers ¶
    - depends_on definition 7.90 The golden ratio ¶ ↺
    - depends_on lemma 7.67 The limit over real exponents ¶
      - depends_on proposition 7.68 The natural base ¶
    - depends_on proposition 7.68 The natural base ¶ ↺
    - depends_on proposition 7.91 The golden ratio ¶ ↺
    - depends_on proposition 7.89 Irrationality of square roots ¶
      - depends_on proposition 7.91 The golden ratio ¶ ↺
  - depends_on proposition 7.89 Irrationality of square roots ¶ ↺
- depends_on lemma 7.67 The limit over real exponents ¶ ↺
- depends_on proposition 7.68 The natural base ¶ ↺
- depends_on proposition 7.63 Laws of real powers ¶ ↺
depends_on proposition 9.141 First properties ¶ ↺
depends_on proposition 9.19 Separation of variables ¶

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`	→	Inverse function	declared	parts/02-mathematical-methods/05-real-analysis.tex:625
`depends_on`	→	Algebra of limits	declared	parts/02-mathematical-methods/05-real-analysis.tex:625
`depends_on`	→	Intermediate value theorem	declared	parts/02-mathematical-methods/05-real-analysis.tex:625
`depends_on`	←	Amplitude and the Jacobi elliptic functions	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5330
`depends_on`	←	The logarithm	declared	parts/02-mathematical-methods/05-real-analysis.tex:1397
`depends_on`	←	First properties	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:5359
`depends_on`	←	Separation of variables	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:656
`proves`	←	ch:05-real-analysis@proof-16	declared	parts/02-mathematical-methods/05-real-analysis.tex:628