The chain rule without rushing

Outside derivative, evaluated at the inside, times the inside derivative.

Learning goals

Differentiate a composition such as sin(3x²) or e^{5x}.
Spot a missing inner derivative in a common error.

Explanation

If f(x) = g(u(x)), then f′(x) = g′(u(x)) · u′(x). The first factor is “how the outer map changes with its input.” The second is “how that input is changing with x.” Forgetting the second factor is the most common slip in the whole course.

Nested compositions need more than one chain. Work from the outside in, and write an intermediate if the expression is crowded. Units still multiply: if the outer function outputs liters and the inner output is temperature, the chain has to carry both rates.

Key terms

Chain rule. The derivative rule for a composition: outer derivative at the inner function, times the inner derivative.
Composite function. A function formed by feeding one function’s output into another function.

Common mistakes

Differentiating the outside and ignoring the inside’s derivative.
Treating sin(x²) as (sin x)² when applying the chain.

Practice

Original Marlow Works items. Check the answer explanation after you try.

1 of 2Limits and Continuity · easy · multiple choice

What is lim (x→3) of (x² − 9) / (x − 3)?

Original Marlow Works item — not a College Board question.

Take your time—this is practice, not a test.

Answer explanation

Differentiate y = √(3x + 2) using the chain rule.

Answer. y′ = (1/2)(3x + 2)^{−1/2} · 3 = 3 / (2√(3x + 2)).

The power −1/2 comes from the outer square root; the 3 is the inner derivative.

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Related resources

External links with reuse status. Marlow Works is independent and does not copy restricted exam or textbook material.

Reference only — do not copy
OpenStax College Algebra 2e
Algebra reference; current terms include noncommercial and AI-ingestion limits.
OpenStax · Restricted — see publisher page · accessed 2026-10-01
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