Calculus I · 2A · lesson
A Visual and Verbal Map of the Chain Rule
Learn a visual and verbal map of the chain rule with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Section overview
The chain rule and compositionsWhat this section is building
Learn a visual and verbal map of the chain rule with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
A small input change passes through a sequence of machines; the total response multiplies the response at each stage.
List the layers, differentiate one layer at a time, and stop only when every input-dependent layer contributes.
Differentiating the outside and leaving the inside unchanged without its derivative factor.
Learning objectives
Identify outer and inner functions in several notational forms and predict the chain-rule factors before calculating.
Follow the Dependency, Not the Typography
Rates multiply because each stage converts units
If pressure changes voltage at volts per kilopascal and voltage changes a reading at display units per volt, then pressure changes the reading at display units per kilopascal. The intermediate volts cancel in the units just as cancels in .
Before the formulas
Composition is the hidden architecture of A Visual and Verbal Map of the Chain Rule. An expression may look like one formula while actually containing several function machines. Draw or list the stages before differentiating. This is especially important when product, quotient, and chain rules appear together, because the outer algebraic connection and the inner compositions must both be respected.
Do not simplify away useful structure too early. A factored derivative often shows each chain factor more clearly and is easier to verify. Simplify after the calculus unless rewriting first genuinely reduces the number of rules required.
Name the layers before touching the derivative
A nested formula becomes manageable when you label its layers. In , the outer operation is "raise to the fifth power," inside that is "add one," inside that is sine, and inside sine is multiplication by three.
Differentiation walks back through those layers from outside to inside. Each layer contributes one derivative factor. Writing the layer map first prevents missing an inner factor, the most common chain-rule error.
A composite function is a dependency chain. If and , then changing changes , and changing changes . The total response is the product of those two local responses:
The same structure may be written as , , , or . Typography changes; dependency does not.
Say the layers aloud
For , the layers are: cosine of a cube of a quadratic-plus-one. The derivative therefore needs three factors: derivative of cosine, derivative of the cube, and derivative of the quadratic-plus-one.
chain-map-01How many nonconstant chain-rule layers appear in ?
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Count each nonconstant function machine from outside to inside.
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