The chain rule is the most-used rule in differentiation and the most-missed. Three errors account for almost all of it.
1. Forgetting the inner derivative
The derivative of sin(3x) is 3cos(3x), not cos(3x). The factor of three is the whole content of
the rule. On a graph the mistake is obvious: the wrong answer has the right shape at a third of the
height.
2. Changing the inside
The derivative of sin(3x) starts cos(3x) — the argument does not move. Writing cos(x) treats the
composition as though it had already been unwrapped.
3. Stopping after one layer
sqrt(sin(x²)) has three layers, so three factors appear. Peeling one and stopping leaves an answer
that is wrong by a whole function.
Each of these is visible in the chain rule worked examples, where the graph of the correct derivative is drawn against the function itself.