Unit 3.1 – The Chain Rule (Composite Functions)

AP® Calculus AB & BC | Differentiating Composite Functions

The Chain Rule is the essential technique for differentiating composite functions—functions "inside" other functions. Anytime you see nested functions (e.g. \( \sin (3x) \), \( e^{x^2} \), \( \ln(5x + 1) \)), the chain rule lets you compute derivatives quickly and accurately. It's a must for AP® Calculus MCQ and FRQs!

🌟 Chain Rule Formula (Must Memorize!)

Formal Chain Rule (Composite Function \( y = f(g(x)) \))

\[ \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) \]

Or Leibniz form: if \( y = f(u) \) and \( u = g(x) \), then:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

How to Use the Chain Rule:

Identify the outer function and inner function.
Take derivative of outer function (leave inner untouched).
Multiply by derivative of inner function.
Simplify! Repeat for multiple layers of composition.

📐 Generalizations & Alternate Forms

Multiple Function Compositions (Nested Chain Rule)

\[ \frac{d}{dx} [f(g(h(x)))] = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x) \]

Keep "peeling" the functions: always start outermost, work inward!

Quick Rewrites for Powers, Roots & Reciprocals:

\( \frac{d}{dx} [(g(x))^n] = n \cdot [g(x)]^{n-1} \cdot g'(x) \)
\( \frac{d}{dx}[\sqrt{g(x)}] = \frac{g'(x)}{2\sqrt{g(x)}} \)
\( \frac{d}{dx}[1/g(x)] = -\frac{g'(x)}{[g(x)]^2} \)

💡 Chain Rule Shortcuts & Tricks

CUE: Chain Rule is needed anywhere you see a function inside another function.
Classic AP® forms: \( \sin(ax) \), \( \cos(bx+c) \), \( e^{h(x)} \), \( \ln(ax + b) \), \( (something)^n \).
Rewrite roots & fractions as powers: \( \sqrt{x} = x^{1/2} \), \(1/x = x^{-1}\).
Implicit Differentiation: Use chain rule for derivatives involving \(y\) as a function of \(x\): when differentiating \(y^n\), result is \(n y^{n-1} \frac{dy}{dx} \).
For multiple compositions, apply chain rule successively.

📖 Worked Examples

Example 1: \( \frac{d}{dx}[\sin(5x)] \)

Outer: \( \sin(u)\) → Derivative: \(\cos(u)\)
Inner: \(u = 5x\) → Derivative: \(5\)
\( \Rightarrow \cos(5x) \cdot 5 = 5\cos(5x) \)

Example 2: \( \frac{d}{dx}[e^{x^2}] \)

Outer: \(f(u) = e^u\), Derivative: \(e^u\)
Inner: \(u = x^2\), Derivative: \(2x\)
\( \Rightarrow e^{x^2} \cdot 2x = 2x e^{x^2} \)

Example 3: \( \frac{d}{dx}[\sqrt{3x^2 + 1}] \)

Rewrite as \( [3x^2 + 1]^{1/2} \)
Outer: \(u^{1/2}\), Derivative: \(\frac{1}{2}u^{-1/2}\)
Inner: \(u = 3x^2 + 1\), Derivative: \(6x\)
\( \Rightarrow \frac{1}{2}[3x^2 + 1]^{-1/2} \cdot 6x = \frac{6x}{2\sqrt{3x^2+1}} = \frac{3x}{\sqrt{3x^2+1}} \)

Example 4: \( \frac{d}{dx}[\ln(2x + 1)] \)

Outer: \(\ln(u)\), Derivative: \(1/u\)
Inner: \(u = 2x+1\), Derivative: \(2\)
\( \Rightarrow \frac{1}{2x+1} \cdot 2 = \frac{2}{2x+1} \)

Example 5: \( \frac{d}{dx}[\tan(3x^2)] \)

Outer: \(\tan(u)\), Derivative: \(\sec^2(u)\)
Inner: \(u = 3x^2\), Derivative: \(6x\)
\( \Rightarrow \sec^2(3x^2) \cdot 6x = 6x \sec^2(3x^2) \)

❌ Common Mistakes to Avoid

Don't forget to multiply by inner function's derivative! Many AP® errors = missing \(g'(x)\).
Always differentiate outer function **first**, leave the inner intact—then multiply by \(g'(x)\).
Don't treat \(\sqrt{g(x)}\) or \(1/g(x)\) as not requiring chain rule.
For trig, never ignore the "inside"—\(\sin(ax+b) \rightarrow \cos(ax+b)\cdot a\).
For \(e^{h(x)}\), always multiply by \(h'(x)\) at the end.
For multiple layers, repeat the process!

✨ Chain Rule Memory Tricks & Mnemonics

Say it out loud every time: "Derive the outside, leave the inside unchanged, times the derivative of the inside"
Mnemonic song: "Outer prime at inner, times inner prime"
Diagram: Draw arrows to visualize the "chain" as function composition
For \(y = f(g(x))\), always end with \( \cdot g'(x) \)
Picture nested boxes—open outer box first!

📝 Practice Problems

Try These Yourself:

\( \frac{d}{dx}\left[\cos(2x^2+3)\right] \)
\( \frac{d}{dx}\left[e^{5x^3}\right] \)
\( \frac{d}{dx}\left[(x^2+1)^{4}\right] \)
\( \frac{d}{dx}\left[\ln(4x^2)\right] \)
\( \frac{d}{dx}\left[\sin(x^4)\right] \)
\( \frac{d}{dx}\left[\sqrt{7x+2}\right] \)

Answers:

\( -\sin(2x^2+3) \cdot 4x \)
\( e^{5x^3} \cdot 15x^2 \)
\( 4(x^2+1)^3 \cdot 2x = 8x(x^2+1)^3 \)
\( \frac{1}{4x^2} \cdot 8x = \frac{8x}{4x^2} = \frac{2}{x} \)
\( \cos(x^4) \cdot 4x^3 \)
\( \frac{1}{2\sqrt{7x+2}} \cdot 7 = \frac{7}{2\sqrt{7x+2}} \)

✏️ AP® Exam Success – Chain Rule Tips

Justify usage: Always state "By the Chain Rule..." in FRQs
Show layers for full points: write each part before multiplying
Watch out for "product of chains"—combine Chain Rule with Product/Quotient Rules when functions are multiplied/divided
Implicit differentiation uses Chain Rule on every \(y\) term!
Chain Rule may be repeated for "nested" compositions on harder AP® problems