The Chain Rule, Explained Without the Confusion
Updated 2026-07-12
The chain rule differentiates a composed function, one function inside another. It says the derivative of f(g(x)) equals f'(g(x)) times g'(x): differentiate the outer function while keeping the inner one unchanged, then multiply by the derivative of the inner.
The chain rule trips people up because it has two moving parts. The fix is to name them out loud every time: an outer function and an inner function. You differentiate the outer, leave the inner alone, then multiply by the inner's derivative.
Outer and inner
In sin(3x), the outer function is sine and the inner function is 3x. The chain rule says: derivative of the outer (cosine) evaluated at the inner (3x), times the derivative of the inner (3). That gives 3 cos(3x).
The step everyone forgets is the last multiply. Without it you have not used the chain rule at all.
Why it works
A composed function stretches the input twice: once by the inner function, then again by the outer. The chain rule multiplies those two rates of stretching together, which is why you multiply the two derivatives.
Worked example
Differentiate f(x) = (3x + 1)^5.
- Outer function is (something)^5, inner function is 3x + 1.
- Differentiate the outer with the power rule, keeping the inner: 5(3x + 1)^4.
- Multiply by the derivative of the inner, which is 3.
Answer: f'(x) = 15(3x + 1)^4.
The common mistake
Wrong: Writing the derivative of (3x+1)^5 as 5(3x+1)^4 and stopping.
Fix: You still have to multiply by the inner derivative, 3. Forgetting that final multiply is the single most common chain-rule error.
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Frequently asked questions
What is the chain rule in simple terms?
It is the rule for differentiating a function inside another function. Differentiate the outer function, leave the inner one unchanged, then multiply by the derivative of the inner function.
When do you use the chain rule?
Whenever you have a composite function, one expression tucked inside another, like sin(2x), (x^2+1)^3, or e^(5x). If there is an inside function, you need the chain rule.