first derivative: meaning, definition, pronunciation and examples
C2Technical/Academic
Quick answer
What does “first derivative” mean?
In calculus, the rate of change of a function at a given point.
Audio
Pronunciation
Definition
Meaning and Definition
In calculus, the rate of change of a function at a given point; a measure of instantaneous change or slope.
The result of the mathematical operation of differentiation. By extension, it can metaphorically refer to the first major consequence or immediate result of a change in any system or situation.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or semantic differences. Pronunciations may vary slightly.
Connotations
Identical technical connotations. In metaphorical use, it remains a highly specialised borrowing from mathematics.
Frequency
Equally frequent in academic/technical contexts in both varieties. Virtually absent in general discourse.
Grammar
How to Use “first derivative” in a Sentence
the first derivative of [NOUN PHRASE]to find/take/calculate the first derivative[FUNCTION]'s first derivativeVocabulary
Collocations
Examples
Examples of “first derivative” in a Sentence
verb
British English
- To solve this, you must first-derivative the function with respect to t. (Note: highly technical/verbing)
American English
- The software can automatically first-derivative the data set. (Note: highly technical/verbing)
adverb
British English
- [Standard adverbial use is non-existent.]
American English
- [Standard adverbial use is non-existent.]
adjective
British English
- The first-derivative test is used to classify critical points.
American English
- We need the first-derivative information for the optimization algorithm.
Usage
Meaning in Context
Business
Extremely rare. Potentially in quantitative finance or advanced economic modelling.
Academic
Core term in mathematics, physics, engineering, and any field using calculus.
Everyday
Virtually never used.
Technical
Fundamental concept in calculus, essential for describing motion, growth, optimization, and change.
Vocabulary
Synonyms of “first derivative”
Strong
Neutral
Weak
Vocabulary
Antonyms of “first derivative”
Watch out
Common Mistakes When Using “first derivative”
- Using 'derivation' instead of 'derivative' (a derivation is the process, a derivative is the result).
- Misplacing the apostrophe: 'the functions first derivative' should be 'the function's first derivative'.
- Forgetting 'first' when context implies it, leading to ambiguity with higher-order derivatives.
FAQ
Frequently Asked Questions
Often, yes. 'Derivative' typically means the first derivative. However, specifying 'first' is crucial when discussing higher-order derivatives (second, third, etc.) to avoid ambiguity.
It is almost exclusively a mathematical term. Its use in economics, finance, or social sciences is a direct metaphor borrowed from calculus and remains technical.
Slope is a static property of a line. The first derivative is the function that gives the slope of the tangent line (the instantaneous slope) at every point on a curve.
Common notations include: f'(x) (Lagrange's), dy/dx (Leibniz's), or Df(x). For the derivative at a specific point a: f'(a) or dy/dx |_{x=a}.
In calculus, the rate of change of a function at a given point.
First derivative is usually technical/academic in register.
First derivative: in British English it is pronounced /ˌfɜːst dɪˈrɪv.ə.tɪv/, and in American English it is pronounced /ˌfɝːst dɪˈrɪv.ə.t̬ɪv/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “[None directly. Metaphorical: 'the first derivative of the policy change was increased public unrest.']”
Learning
Memory Aids
Mnemonic
Think of a car's speedometer. The speed is the FIRST DERIVATIVE of the car's position (the odometer reading) with respect to time. Speed = derivative of position.
Conceptual Metaphor
CHANGE IS MOTION / A QUANTITY IS A PATH. The first derivative is the instantaneous 'speed' or 'steepness' of that path.
Practice
Quiz
In a velocity-time graph, what does the first derivative of velocity represent?