Take \(y=e^{\sin(3x)}\). The layer inventory from the previous chapter gives three layers, so the derivative must have three factors.
Layer inventory.
- Outermost: \(e^{\square}\)
- Middle: \(\sin(\square)\)
- Innermost: \(3x\)
Fill the factors in order, outside in.
- Derivative of \(e^{\square}\) at the middle layer: \(e^{\sin(3x)}\)
- Derivative of \(\sin(\square)\) at the innermost layer: \(\cos(3x)\)
- Derivative of \(3x\): \(3\)
Full product.
\[\frac{dy}{dx}=e^{\sin(3x)}\cdot\cos(3x)\cdot 3\]
The factor-count check. Count the factors you wrote: three. Count the layers you inventoried: three. They match, so no layer was skipped. If you had written only \(e^{\sin(3x)}\cos(3x)\), the count would be two against three layers, and the missing factor would be immediately visible.
Where the dropped layer usually hides. In a three-layer composite the middle factor is the one most often lost, because it is neither the obvious outer function nor the final linear piece. The count check catches it without requiring you to re-derive anything.