Derivative: Meaning, Notation, and Example
The instantaneous rate of change of a function.
Definition and meaning
A derivative is the limit of average rates over shrinking intervals. It is also the slope of the tangent line when that geometric interpretation applies.
The term derivative names a precise mathematical idea, not merely a visual pattern. Read its notation as a statement with hypotheses, quantities, and a conclusion.
Notation
Read every symbol with its domain and hypotheses; notation compresses an argument but does not remove its conditions.
Long description
Read the diagram from top to bottom. Each numbered box names one decision or mathematical operation. Arrows show the required order; the text labels remain the complete interpretation in print, dark mode, and nonvisual reading.
Worked example
Identify the quantities in the notation, state the relevant hypothesis, and explain what conclusion the derivative statement permits.
Common confusion
Do not treat derivative as a keyword that automatically selects a formula; its hypotheses and domain still control the conclusion.