Calculus I · 2A · lesson
Combining Product, Quotient, and Chain Rules
Learn combining product, quotient, and chain rules with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Section overview
The chain rule and compositionsWhat this section is building
Learn combining product, quotient, and chain rules 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
Differentiate expressions whose structure requires multiple rules in a controlled order.
Expressions That Need Several Rules
Before the formulas
In Combining Product, Quotient, and Chain Rules, the phrase "outside to inside" is useful only when paired with structure. Keep the inner expression unchanged while differentiating the outer layer, then multiply by the derivative of that inner expression. If the inner expression is itself composite, continue inward.
A missing inner derivative is the signature chain-rule error. Check units or scaling to catch it. If an inner quantity changes three times as fast, the final output rate should reflect that factor of three. The chain rule is a rate-conversion law, not punctuation attached to parentheses.
Complex derivatives are built from a small rule set
A long expression may combine products, quotients, and compositions, but it does not require a new formula. Mark the outer structure, apply its rule, and then handle each inner derivative recursively.
Do not simplify too early. A factored derivative often displays the method more clearly, reduces algebraic risk, and is easier to check against the original function's domain.
Real models seldom volunteer to use exactly one differentiation rule. A damped vibration may be an exponential times a sine; a normalized response may be a quotient containing a composition. The goal is not to choose one rule but to organize several rules without losing the structure.
Work from the largest operation inward, keep intermediate expressions factored, and postpone cosmetic simplification. A correct unsimplified derivative is more valuable than a beautifully simplified wrong one.
For
the outermost operation is multiplication, so use the product rule. The derivative of the sine factor then requires the chain rule.
Product rule outside, chain rule inside
Differentiate
Worked solution
Write a real attempt before opening the supplied answer.
Quotient rule with two chain-rule derivatives
For
Order of operations for derivatives
Identify the outermost operation first. Apply its rule, then differentiate each component using whatever inner rules it requires. Do not begin at the deepest parentheses and lose track of product or quotient structure surrounding them.
Differentiate .
Differentiate .
Differentiate .
Differentiate .
A damped vibration
A machine displacement is modeled by
The velocity is
The product rule accounts for shrinking amplitude and oscillation; the chain rule supplies both inner factors and .
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