Calculus I · Unit 2A · lesson

Combining Product, Quotient, and Chain Rules

Concept

Learning objectives

Differentiate expressions whose structure requires multiple rules in a controlled order.

Expressions That Need Several Rules

Explanation

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.

Explanation

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

y=x^2\sin(3x),

the outermost operation is multiplication, so use the product rule. The derivative of the sine factor then requires the chain rule.

Guided walkthrough

Product rule outside, chain rule inside

Differentiate

y=x^2e^{3x}.

Answer reveal

Worked solution

Your setup or answer

Write a real attempt before opening the supplied answer.

Worked example

Quotient rule with two chain-rule derivatives

For

y=\frac{\sin(x^2)}{e^{x}},

\boxed{y'=\frac{2x\cos(x^2)e^x-\sin(x^2)e^x}{e^{2x}}} =\boxed{e^{-x}[2x\cos(x^2)-\sin(x^2)]}.

Decision

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.

Exercise

Differentiate $(x^2+1)^4\sin x$ .

Exercise

Differentiate $e^{x^2}/(x+1)$ .

Exercise

Differentiate $\ln(x^2+1)\cos(2x)$ .

Exercise

Differentiate $[\tan(3x)]^5$ .

Modeling lab

A damped vibration

A machine displacement is modeled by

x(t)=5e^{-0.2t}\cos(4t).

The velocity is

\begin{aligned} x'(t) &=5\left[-0.2e^{-0.2t}\cos(4t)+e^{-0.2t}(-4\sin(4t))\right]\\ &=e^{-0.2t}\left[-\cos(4t)-20\sin(4t)\right]. \end{aligned}

The product rule accounts for shrinking amplitude and oscillation; the chain rule supplies both inner factors $-0.2$ and $4$ .

After the explanation

Use the section idea

Reading lens

Read nested functions from the outside inward, but multiply local response factors through every layer.

Mental model

A small input change passes through a sequence of machines; the total response multiplies the response at each stage.

Decision

List the layers, differentiate one layer at a time, and stop only when every input-dependent layer contributes.

Common trap

Differentiating the outside and leaving the inside unchanged without its derivative factor.

Check yourself

Can you annotate every factor in your derivative with the layer that produced it?

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