Here's the standard textbook statement of the chain rule: "If y = f(g(x)), then dy/dx = f'(g(x)) · g'(x)." And here's what you actually need to remember: differentiate the outside, leave the inside alone, then multiply by the derivative of the inside.
That's it. That's the entire rule. The rest is bookkeeping.
The pattern
Whenever you see one function stuck inside another — (3x + 1)⁴, sin(x²), e^(2x), ln(cos x) — you've got a chain-rule situation. The outside function is whatever wraps everything else. The inside is whatever's inside the outermost parentheses or function call.
Worked example: (3x + 1)⁴
Outside function: something to the 4th power. Its derivative: 4 times something to the 3rd power. So: 4(3x + 1)³.
Inside function: 3x + 1. Its derivative: 3.
Multiply: 4(3x + 1)³ · 3 = 12(3x + 1)³. Done.
Worked example: sin(x²)
Outside: sine of something. Derivative: cosine of that something. So: cos(x²). Notice the x² stays — you don't differentiate it yet.
Inside: x². Derivative: 2x.
Multiply: cos(x²) · 2x = 2x · cos(x²).
Worked example: ln(cos x) — because it's slippery
Outside: natural log of something. Derivative: 1/something. So: 1/cos(x).
Inside: cos(x). Derivative: −sin(x).
Multiply: (1/cos(x)) · (−sin(x)) = −sin(x)/cos(x) = −tan(x).
When you have three layers
Sometimes functions are nested three deep — for example, sin³(2x), which really means [sin(2x)]³. You peel one layer at a time from the outside in, multiplying each derivative as you go.
Outermost: something cubed → 3[sin(2x)]². Multiply by the derivative of sin(2x), which is itself a chain: cos(2x) · 2. Full answer: 3[sin(2x)]² · cos(2x) · 2 = 6 sin²(2x) · cos(2x).
The common mistakes, in order of how often they happen
- Forgetting to multiply by the inside derivative. You differentiate the outside but then don't chain. This is the single most common error on AP Calc. If your answer doesn't include a factor from the inside function, check.
- Chaining when you shouldn't. 3x is not a chain-rule problem. Its derivative is just 3. Only chain when there's a real "inside" function.
- Differentiating the inside too early. In sin(x²), the x² stays as x² while you differentiate the sine. Don't swap it to 2x until you multiply.
