Chain rule is one of 9 derivatives formulas in the calculus section of this library, and it is used at ap · university level.
Why chain rule works
Rates multiply the way gears do. If the inner expression changes three times as fast as x, and the outer function changes four times as fast as the inner expression, then the composition changes twelve times as fast as x. That is exactly the outer derivative times the inner one.
What each symbol means
$g$ is the inner function and $f$ the outer function.
Chain rule: when it holds
Both required derivatives must exist.
When it stops applying
If the inner function has no derivative at the point, the product on the right does not exist. With an inner function like |x| at the origin the rule gives nothing, and the composition has to be tested directly against the definition of the derivative.
Chain rule: a worked example
$\frac{d}{dx}(3x+1)^4=4(3x+1)^3(3)=12(3x+1)^3$.
The mistake to avoid
What people do: Writing the derivative of (3x+1) to the fourth as 4(3x+1) cubed and stopping there.
Why it goes wrong: The inside was never differentiated, so the whole answer is short by the factor 3 that the inside contributes.
Do this instead: After taking the outer derivative, always multiply by the derivative of whatever sat inside the parentheses.
Chain rule: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $g$ is the inner function and $f$ the outer function.
- Check the conditions before substituting. Both required derivatives must exist.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Derivatives
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where chain rule comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Derivative Rules — the lesson behind this formula: differentiate sums, products, quotients, and compositions.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about chain rule
How do I spot which part is the inside function?
It is whatever you would type into a calculator first. In sin of x squared you square before you take a sine, so x squared is the inside.
Do I need this rule for plain sin x?
No. The inside is just x, whose derivative is 1, and multiplying by 1 changes nothing, so the basic rule already handles it.
How does it work with three layers?
Peel one layer at a time and multiply everything together. The derivative of sin of (3x+1) to the fourth is cos of that same expression, times 4(3x+1) cubed, times 3.
Why is the outer derivative evaluated at the inner value?
The outer function never sees x. It only ever receives the number the inner function hands it, so its rate has to be measured at that received value.