The layout on trigonometric composites
Trig composites follow the same two-slot procedure. Only the outer derivative changes.
Example 1: \(\sin(4x)\)
- Outer layer: \(\sin(\square)\), whose derivative is \(\cos(\square)\).
- Inner layer: \(4x\), whose derivative is \(4\).
\[\frac{d}{dx}\sin(4x) = \cos(4x) \cdot 4\]
The argument of the cosine is \(4x\), not \(x\). Writing \(\cos(x)\cdot 4\) breaks the rule that the outer factor is evaluated at the inner function.
Example 2: \(\cos(x^3)\)
- Outer layer: \(\cos(\square)\), whose derivative is \(-\sin(\square)\).
- Inner layer: \(x^3\), whose derivative is \(3x^2\).
\[\frac{d}{dx}\cos(x^3) = -\sin(x^3) \cdot 3x^2\]
Two things to watch. First, the minus sign belongs to the outer factor, so it appears before the sine. Second, the inner factor is \(3x^2\), not \(3x\) and not \(2x\).
The pattern
| Composite | Outer factor | Inner factor |
|---|---|---|
| \(\sin(4x)\) | \(\cos(4x)\) | \(4\) |
| \(\cos(x^3)\) | \(-\sin(x^3)\) | \(3x^2\) |
In both rows the outer factor keeps the inner expression intact inside the trig function, and the inner factor is written separately at the end.