Math 2.0

Real functions Theorem

General Leibniz rule

Builds on Higher derivativeProduct rule

Statement

Let nn be a natural number, and let ff and gg be nn times differentiable real functions on an open interval II. Then fgf g is nn times differentiable on II, and (fg)(n)=∑k=0n(nk)f(k)g(n−k).(f g)^{(n)} = \sum_{k=0}^{n} \binom{n}{k} f^{(k)} g^{(n-k)}.

Remarks

The proof is an induction on nn with the product rule, like the proof of the binomial theorem. For n=1n = 1 it is the product rule. For n=2n = 2 it is (fg)′′=f′′g+2f′g′+fg′′(f g)'' = f'' g + 2 f' g' + f g''. Mathlib states it as iteratedDeriv_mul.

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To understand this page from the ground up, read these 10 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 levelDerivativedefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionDifferentiable functions are continuous, statement, no evidence levelDifferentiable functi…statementDerivative function, definition, no evidence levelDerivative functiondefinitionProduct rule, statement, no evidence levelProduct rulestatementHigher derivative, definition, no evidence levelHigher derivativedefinitionGeneral Leibniz rule, statement, no evidence levelGeneral Leibniz 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.
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  1. Limit point of a set definition
  2. Limit of a function definition
  3. Continuity at a point definition
  4. Derivative definition
  5. Differentiable function definition
  6. Differentiable functions are continuous statement
  7. Derivative function definition
  8. Product rule statement
  9. Higher derivative definition
  10. General Leibniz rule statement

Further reading

  1. Web https://en.wikipedia.org/wiki/General_Leibniz_rule, lead section
About this page
Tag0K9QCK6WJC
URIm2://0K9QCK6WJC
Kindstatement, theorem
WritingW1 stated
MSC26A24
StatusThis copy of the site has no build data. Run m2 status 0K9QCK6WJC.
UsesHigher derivative, Product rule
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