The Chain Rule is one of the most important derivative techniques in AP Calculus AB & BC — and one of the most commonly tested. It appears in multiple-choice, free response, and applied problems involving related rates, exponential growth, and composition of functions.
In this RevisionDojo guide, you’ll learn:
- The Chain Rule formula
- How to identify when to use it
- Step-by-step examples
- Common mistakes and how to avoid them
📚 The Chain Rule Formula
If y=f(g(x))y = f(g(x)), then:
dydx=f′(g(x))⋅g′(x)\frac{dy}{dx} = f'(g(x)) \cdot g'(x)
In words: Differentiate the outer function, leave the inner function untouched, then multiply by the derivative of the inner function.
🔍 When to Use the Chain Rule
Look for:
- Functions inside other functions (sin(3x2)\sin(3x^2), eln(x)e^{\ln(x)}, 4x+1\sqrt{4x+1})
- Exponents that are more than just xnx^n
- Trig, log, or exponential functions with a non-xx argument
📝 Step-by-Step Examples
Example 1:
Find ddxsin(3x2)\frac{d}{dx} \sin(3x^2)
- Outer function: sin(u)\sin(u) → derivative is cos(u)\cos(u)
- Inner function: u=3x2u = 3x^2 → derivative is 6x6x
- Apply Chain Rule:
ddxsin(3x2)=cos(3x2)⋅6x\frac{d}{dx} \sin(3x^2) = \cos(3x^2) \cdot 6x
Example 2:
Find ddxe5x+1\frac{d}{dx} e^{5x+1}
- Outer: eue^u → derivative is eue^u
- Inner: u=5x+1u = 5x+1 → derivative is 55
- Apply:
ddxe5x+1=e5x+1⋅5\frac{d}{dx} e^{5x+1} = e^{5x+1} \cdot 5
Example 3:
Find ddx(4x2+1)10\frac{d}{dx} (4x^2+1)^{10}
- Outer: u10u^{10} → derivative is 10u910u^9
- Inner: u=4x2+1u = 4x^2+1 → derivative is 8x8x
- Apply:
ddx(4x2+1)10=10(4x2+1)9⋅8x\frac{d}{dx} (4x^2+1)^{10} = 10(4x^2+1)^9 \cdot 8x
⚠️ Common Chain Rule Mistakes
- Forgetting the inner derivative — the most frequent AP Calculus error.
- Mixing up Product Rule and Chain Rule when a product of functions is also a composition.
- Over-simplifying too soon, leading to algebra mistakes.
📊 Practice Strategy from RevisionDojo
- Drill simple compositions until the process is automatic.
- Practice mixing Chain Rule with Product and Quotient Rules in the same problem.
- Apply it to real-world word problems like related rates.
🧭 Final Advice from RevisionDojo
The Chain Rule is not optional — it’s one of the most tested differentiation techniques in AP Calculus AB & BC.
Master it early, and problems that once looked intimidating will become straightforward step-by-step processes.
