Leibniz rule
E582439
The Leibniz rule is a fundamental property of derivatives stating that the derivative of a product equals the sum of each factor’s derivative times the other factor.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Leibniz rule canonical | 2 |
| Leibniz formula for higher derivatives of products | 1 |
| Leibniz integral rule | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6295503 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Leibniz rule Context triple: [Lie derivative, satisfies, Leibniz rule]
-
A.
Lie derivative
The Lie derivative is a fundamental differential operator in differential geometry that measures how a tensor field changes along the flow generated by a vector field.
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B.
Itô’s lemma
Itô’s lemma is a fundamental result in stochastic calculus that generalizes the chain rule to functions of stochastic processes, especially Brownian motion.
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C.
Lie bracket
The Lie bracket is a bilinear, antisymmetric operation on a Lie algebra that measures the noncommutativity of its elements and encodes its infinitesimal structure.
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D.
Krak de l’Hospital
Krak de l’Hospital is an alternative name for Krak des Chevaliers, the famous medieval Crusader castle in Syria renowned for its massive fortifications and strategic importance.
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E.
Radon–Nikodym derivative
The Radon–Nikodym derivative is a function that represents how one measure changes with respect to another absolutely continuous measure, playing a central role in modern probability theory and measure theory.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Leibniz rule Target entity description: The Leibniz rule is a fundamental property of derivatives stating that the derivative of a product equals the sum of each factor’s derivative times the other factor.
-
A.
Lie derivative
The Lie derivative is a fundamental differential operator in differential geometry that measures how a tensor field changes along the flow generated by a vector field.
-
B.
Itô’s lemma
Itô’s lemma is a fundamental result in stochastic calculus that generalizes the chain rule to functions of stochastic processes, especially Brownian motion.
-
C.
Lie bracket
The Lie bracket is a bilinear, antisymmetric operation on a Lie algebra that measures the noncommutativity of its elements and encodes its infinitesimal structure.
-
D.
Krak de l’Hospital
Krak de l’Hospital is an alternative name for Krak des Chevaliers, the famous medieval Crusader castle in Syria renowned for its massive fortifications and strategic importance.
-
E.
Radon–Nikodym derivative
The Radon–Nikodym derivative is a function that represents how one measure changes with respect to another absolutely continuous measure, playing a central role in modern probability theory and measure theory.
- F. None of above. chosen
Statements (39)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
property of derivatives ⓘ rule in calculus ⓘ |
| appliesTo |
C^n functions
ⓘ
derivative of a product ⓘ differentiable functions ⓘ smooth functions ⓘ |
| category | differentiation rule ⓘ |
| expresses | product rule for derivatives ⓘ |
| field |
calculus
ⓘ
mathematical analysis ⓘ |
| generalizes | product rule for first derivatives ⓘ |
| hasConsequence |
derivative of a constant times a function equals the constant times the derivative of the function
ⓘ
derivative of x^n can be computed by repeated application of the product rule ⓘ |
| hasFormula | (f g)' = f' g + f g' ⓘ |
| hasGeneralForm | d^n(fg)/dx^n = Σ_{k=0}^n (n choose k) f^{(k)} g^{(n-k)} ⓘ |
| hasNotation | (fg)^{(n)} = Σ_{k=0}^n (n choose k) f^{(k)} g^{(n-k)} ⓘ |
| hasVariant |
Leibniz integral rule
ⓘ
linked to:
Leibniz rule
|
| holdsFor |
complex-valued differentiable functions
ⓘ
differentiable vector-valued functions ⓘ differential operators ⓘ real-valued differentiable functions ⓘ |
| involves |
binomial coefficients
ⓘ
higher-order derivatives ⓘ |
| isEquivalentTo | product rule in elementary calculus ⓘ |
| logicalType | universal statement about differentiable functions ⓘ |
| namedAfter | Gottfried Wilhelm Leibniz ⓘ |
| relatedTo |
chain rule
ⓘ
linearity of differentiation ⓘ |
| requires | existence of derivatives of the factors ⓘ |
| states | the derivative of a product equals the sum of each factor’s derivative times the other factor ⓘ |
| usedIn |
Taylor series expansions
ⓘ
differential equations ⓘ distribution theory ⓘ functional analysis ⓘ multivariable calculus ⓘ operator calculus ⓘ |
| usedToProve |
Leibniz formula for higher derivatives of products
ⓘ
linked to:
Leibniz rule
properties of polynomial derivatives ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Leibniz rule Description of subject: The Leibniz rule is a fundamental property of derivatives stating that the derivative of a product equals the sum of each factor’s derivative times the other factor.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Leibniz rule
linked to: Leibniz rule