derivative
Let an agent explain derivatives and compute them, apply derivatives to rates, optimisation and physics, interpret spectral quantities as derivatives, and separate the financial sense.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
Let an agent explain derivatives and compute them, apply derivatives to rates, optimisation and physics, interpret spectral quantities as derivatives, and separate the financial sense.
In calculus, the rate at which a function changes with respect to its variable, defined as the limit of the difference quotient and interpreted as the slope of the tangent or the instantaneous rate of change, with rules for computing derivatives of sums, products, quotients and compositions, and extensions to partial, directional and higher derivatives; in physics and engineering, spectral quantities such as spectral radiance are derivatives with respect to frequency or wavelength.
What it is for: Rates of change of functions.
It can be explain and compute derivatives; apply to rates and optimisation; interpret spectral derivatives; separate other senses.
Distinguishing features
Limit definition
Local rate of change
Rules of differentiation
Generalisable
What it looks like
Not physical; slopes of curves and formulas.
How it is recognised
Rate of change of a function
Slope of the tangent
A financial derivative is a contract, not a rate
Related models
is a kind of - category
is a kind of - category
is related to - Riemann sums and integrals
is related to - derivative of velocity
In practice
Families and kinds
ordinary derivatives
partial and directional derivatives
higher-order derivatives
spectral derivatives such as spectral radiance
derivatives in vector calculus
Standards and regulation
Mathematical notation conventions
Radiometric quantity definitions for spectral derivatives
Failure modes and hazards
Confusing derivative with difference or with financial derivatives
Differentiation rule errors
Misinterpreting spectral quantities per frequency versus per wavelength
Also called
Where this came from
wikidata · CC0 1.0
Also registered as vr.tr.derivative-substance
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Define What a derivative is.
Concepts.
Definition
Limit definition.
Definition
Definition.
- How is the derivative defined as a limit, and how is it interpreted as slope and rate of change? definition
- Is the calculus sense, a financial derivative or a chemical derivative meant? boundary
Rules
Differentiation rules.
Rules
Rules.
- What are the rules for derivatives of sums, products, quotients and compositions? definition
- What is the derivative of this function? action
Apply Using derivatives.
Applications.
Rates
Rates and optimisation.
Rates
Rates.
- How are derivatives used to find rates of change, extrema and related rates in this problem? action
- How is a function analysed using its derivatives? action
Physics
Derivatives in science.
Physics
Science.
- How do velocity, acceleration and other physical quantities arise as derivatives? definition
- What are spectral quantities such as spectral radiance with respect to frequency, and why do they differ per wavelength? definition
Extend Beyond one variable.
Advanced.
Partial
Partial and directional derivatives.
Partial
Partial derivatives.
- What are partial, directional and total derivatives, and how are gradients used? definition
- How are they computed for this function? action
Numeric
Numerical differentiation.
Numeric
Numerical.
- How are derivatives approximated numerically and computed by automatic differentiation? definition
- What errors arise? boundary
Learn Teaching and history.
Education.
Teach
Teaching derivatives.
Teach
Teaching.
- How can derivatives be taught with graphs and motion examples? action
- Which misconceptions arise? provenance
History
History.
History
History.
- How did Newton, Leibniz and later mathematicians develop the derivative? provenance
- Which references are standard? provenance
What the second pass must settle
- Should differentiation rules be a separate entry?
- How should calculus resources be linked?
- How should radiometric definitions be linked?