Math 2.0

Real functions Differentiation Theorem

Derivative of an inverse function

Builds on Continuity of the inverse of a strictly monotone continuous functionDerivativeChain rule

Statement

Let II be an open interval and let f ⁣:I→ℝf \colon I \to \mathbb{R} be continuous and strictly monotone. Assume that ff is differentiable at a∈Ia \in I with f′(a)≠0f'(a) \ne 0, and let b=f(a)b = f(a). Then f−1f^{-1} is differentiable at bb, and (f−1)′(b)=1f′(a)=1f′(f−1(b)).(f^{-1})'(b) = \frac{1}{f'(a)} = \frac{1}{f'(f^{-1}(b))}.

Remarks

The inverse is continuous, so f−1(b+k)→af^{-1}(b + k) \to a as k→0k \to 0. The difference quotient of f−1f^{-1} at bb is the reciprocal of a difference quotient of ff at aa. If f′(a)=0f'(a) = 0, then f−1f^{-1} is not differentiable at bb. An example is f(x)=x3f(x) = x^3 at a=0a = 0. For example, it gives the derivative of the logarithm from the derivative of the exponential function.

Reading path

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionStrictly monotone function, definition, no evidence levelStrictly monotone fun…definitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionOne-sided limit, definition, no evidence levelOne-sided limitdefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionEpsilon-delta definition of continuity, statement, no evidence levelEpsilon-delta definit…statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementIntermediate value theorem, statement, no evidence levelIntermediate value th…statementComposition of continuous functions is continuous, statement, no evidence levelComposition of contin…statementOne-sided limits of a monotone function exist, statement, no evidence levelOne-sided limits of a…statementChain rule, statement, no evidence levelChain rulestatementContinuity of the inverse of a strictly monotone continuous function, statement, no evidence levelContinuity of the inv…statementDerivative of an inverse function, statement, no evidence levelDerivative of an inve…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Limit point of a set definition
  2. Sequence definition
  3. Strictly monotone function definition
  4. Limit of a function definition
  5. Real sequence definition
  6. Continuity at a point definition
  7. Derivative definition
  8. One-sided limit definition
  9. Bounded sequence definition
  10. Limit of a sequence definition
  11. Monotone sequence definition
  12. Continuous function definition
  13. Epsilon-delta definition of continuity statement
  14. Monotone convergence theorem for sequences statement
  15. Intermediate value theorem statement
  16. Composition of continuous functions is continuous statement
  17. One-sided limits of a monotone function exist statement
  18. Chain rule statement
  19. Continuity of the inverse of a strictly monotone continuous function statement
  20. Derivative of an inverse function statement

Where this leads

16 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Inverse_function_rule, Derivation
About this page
Tag0M0RZ3HSB0
URIm2://0M0RZ3HSB0
Aliasesinverse-function-rule
Kindstatement, theorem
WritingW1 stated
MSC26A24
StatusThis copy of the site has no build data. Run m2 status 0M0RZ3HSB0.
UsesContinuity of the inverse of a strictly monotone continuous function, Derivative, Chain rule
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