second derivative: meaning, definition, pronunciation and examples
C1Technical / Academic
Quick answer
What does “second derivative” mean?
A calculus operator measuring the rate of change of a rate of change (the derivative of a derivative). In physical contexts, it describes acceleration when applied to position.
Audio
Pronunciation
Definition
Meaning and Definition
A calculus operator measuring the rate of change of a rate of change (the derivative of a derivative). In physical contexts, it describes acceleration when applied to position.
Metaphorically, it refers to a change in the rate of change of any quantity, describing acceleration, concavity, or inflection in graphs. In finance, it might describe the rate at which a risk metric changes.
Dialectal Variation
British vs American Usage
Differences
No significant semantic or usage differences exist. Spelling of related terms (e.g., 'behaviour' vs. 'behavior') may apply in context.
Connotations
Identical technical connotation in both dialects.
Frequency
Equally frequent in formal academic STEM contexts in both regions. Rare to non-existent in everyday speech.
Grammar
How to Use “second derivative” in a Sentence
The second derivative of [FUNCTION/VARIABLE]Take the second derivative with respect to [VARIABLE][FUNCTION] has a positive second derivativeVocabulary
Collocations
Examples
Examples of “second derivative” in a Sentence
verb
British English
- We need to **second-differentiate** the expression to find the point of inflection.
American English
- You must **take the second derivative** of the position function to get acceleration.
adverb
British English
- [No standard adverbial form for this noun phrase]
American English
- [No standard adverbial form for this noun phrase]
adjective
British English
- The **second-derivative** test confirmed it was a local maximum.
American English
- The **second-derivative** information is crucial for the optimisation algorithm.
Usage
Meaning in Context
Business
Rare; only in highly quantitative finance or economics modelling.
Academic
Core term in calculus, physics, engineering, and any mathematical science.
Everyday
Virtually never used. A layperson might say 'how quickly the speed is changing' instead.
Technical
The primary domain. Used precisely to analyze curvature, optimization, and motion.
Vocabulary
Synonyms of “second derivative”
Strong
Neutral
Weak
Vocabulary
Antonyms of “second derivative”
Watch out
Common Mistakes When Using “second derivative”
- Using 'second derivation' (incorrect noun form).
- Misplacing the prime symbol: writing f''(x) as f'2(x).
- Confusing the second derivative test for extrema with the first derivative test.
FAQ
Frequently Asked Questions
Yes, exactly. It is the derivative of the first derivative function.
Common notations are f''(x), d²y/dx², or y''.
Yes. A zero second derivative at a point may indicate a point of inflection, where the concavity of the graph changes.
It is used to calculate acceleration in physics, determine optimal production levels in economics (via the second derivative test), and analyse curvature in engineering design.
A calculus operator measuring the rate of change of a rate of change (the derivative of a derivative). In physical contexts, it describes acceleration when applied to position.
Second derivative is usually technical / academic in register.
Second derivative: in British English it is pronounced /ˌsɛk.ənd dɪˈrɪv.ə.tɪv/, and in American English it is pronounced /ˌsɛk.ənd dəˈrɪv.ə.ṭɪv/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “[None directly; term is purely technical]”
Learning
Memory Aids
Mnemonic
Think of a speedometer (first derivative of position). The second derivative is how quickly the needle on that speedometer is moving.
Conceptual Metaphor
CHANGE OF CHANGE IS ACCELERATION / CURVATURE.
Practice
Quiz
What does a positive second derivative of a function at a point typically indicate about its graph?