Chain Rule: Meaning, Notation, and Example
Differentiates a composition from outside to inside.
Definition and meaning
Differentiate the outer function at the unchanged inner expression, then multiply by the inner derivative.
The term chain rule 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 chain rule statement permits.
Common confusion
Do not treat chain rule as a keyword that automatically selects a formula; its hypotheses and domain still control the conclusion.