Math 2.0

Real functions Differentiation Theorem

Chain rule

Builds on DerivativeComposition of continuous functions is continuous

Statement

Let ff be a real function on an open interval II that is differentiable at a∈Ia \in I. Let gg be a real function on an open interval JJ with f(I)⊆Jf(I) \subseteq J, and assume that gg is differentiable at f(a)f(a). Then g∘fg \circ f is differentiable at aa, and (g∘f)′(a)=g′(f(a)) f′(a).(g \circ f)'(a) = g'(f(a)) \, f'(a).

Remarks

A naive proof multiplies and divides by f(a+h)−f(a)f(a + h) - f(a). This fails when f(a+h)=f(a)f(a + h) = f(a) for small hh. A correct proof uses the linear approximation of gg at f(a)f(a). For example, (sin⁡(x2))′=2xcos⁡(x2)(\sin(x^2))' = 2 x \cos(x^2). In Leibniz notation, dzdx=dzdy⋅dydx\frac{dz}{dx} = \frac{dz}{dy} \cdot \frac{dy}{dx}. Mathlib states the rule as HasDerivAt.comp.

History

Gottfried Wilhelm Leibniz seems to have used the rule first, in a memoir of 1676.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionEpsilon-delta definition of continuity, statement, no evidence levelEpsilon-delta definit…statementComposition of continuous functions is continuous, statement, no evidence levelComposition of contin…statementChain rule, statement, no evidence levelChain rulestatement
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. Limit of a function definition
  3. Continuity at a point definition
  4. Derivative definition
  5. Continuous function definition
  6. Epsilon-delta definition of continuity statement
  7. Composition of continuous functions is continuous statement
  8. Chain rule statement

Where this leads

21 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Chain_rule, Statement
About this page
Tag0RXX2PT1MQ
URIm2://0RXX2PT1MQ
Kindstatement, theorem
WritingW1 stated
MSC26A24
StatusThis copy of the site has no build data. Run m2 status 0RXX2PT1MQ.
UsesDerivative, Composition of continuous functions is continuous
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