Chain Rule

Learn chain rule, multiple layers of composition.

Lesson 2 of 32 parts Practice Flashcards

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Chain Rule

Objective: Explain chain rule, apply the stated method, and check its conditions.

For a composition f(g(x)), the derivative is f′(g(x))g′(x). Differentiate the outer rule while retaining its input, then multiply by the inner derivative. Both component derivatives must exist at the relevant inputs. Do not omit the inner derivative.

Formulas

$(f∘g)′(x)=f′(g(x))g′(x)$

Worked examples

Try itFor f(x)=(6x+7)², find f′(7).Show answer

Step 1: Differentiate the outer square and multiply by the inner derivative. Step 2: f′(x)=2(6x+7)·6. Step 3: f′(7)=588.

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Chain Rule

Objective: Explain chain rule, apply the stated method, and check its conditions.

Formulas

$(f∘g)′(x)=f′(g(x))g′(x)$

Worked examples

Try itFor f(x)=(6x+7)², find f′(7).Show answer

Step 1: Differentiate the outer square and multiply by the inner derivative. Step 2: f′(x)=2(6x+7)·6. Step 3: f′(7)=588.

Multiple Layers of Composition

Objective: Explain multiple layers of composition, apply the stated method, and check its conditions.

A nested power such as (g(x))³ requires the outer power derivative 3g(x)² and the inner derivative g′(x). Repeated layers require repeated chain factors. Parentheses clarify which entire expression receives each power. Keep the whole inner expression inside the remaining power.

Formulas

$d(g(x)³)/dx=3g(x)²g′(x)$

Worked examples

Try itFor f(x)=(x²+7)³, find f′(7).Show answer

Step 1: Use chain rule: 3(x²+b)² times 2x. Step 2: f′(x)=6x(x²+7)². Step 3: At x=7, the value is 131712.

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