The setup
Suppose a solution claims
\[\frac{d}{dx}\left[\sin\left(x^2\right)\right]=\cos\left(x^2\right)\]
This is a plausible-looking answer, and it is wrong. The layout tells you exactly where it broke.
Step 1: layer count
The expression \(\sin(x^2)\) has two layers:
- outer: \(\sin(;)\)
- inner: \(x^2\)
Two layers means two factors. The claimed answer has one factor, \(\cos(x^2)\). The layer count and the factor count disagree, so the solution is already flagged.
Step 2: factor count
Write the block properly:
\[\frac{d}{dx}\left[\sin\left(x^2\right)\right]=\cos\left(x^2\right)\cdot 2x\]
The correct answer has two factors. The wrong answer is missing the innermost layer derivative, \(2x\).
Step 3: inner-function preservation
Check that the inner expression was copied unchanged into the outer derivative factor. Here \(\cos(x^2)\) does keep \(x^2\) intact, so this check passes. The error is not a drifted argument; it is a dropped factor.
A second failure mode
Compare with
\[\frac{d}{dx}\left[\ln\left(x^2\right)\right]=\frac{1}{x}\cdot 2x=2\]
Here the factor count is right and the innermost derivative is right, but the outer derivative was evaluated at \(x\) instead of at the inner function. The correct block is
\[\frac{d}{dx}\left[\ln\left(x^2\right)\right]=\frac{1}{x^2}\cdot 2x=\frac{2}{x}\]
The inner-function-preservation check is the one that catches this, and it is the only check that catches it.
Why three checks
Each check targets a different failure mode:
| Check | Failure it catches |
|---|---|
| Layer count | Wrong number of factors expected |
| Factor count | A dropped or duplicated factor |
| Inner-function preservation | Outer derivative evaluated at the wrong argument |
Running them in this order means the first failing check names the broken step.