The last-operation test works the same way across every function family you meet in AP Calculus BC. What changes is only the shape of the wrapper.
| Expression | Outer function | Inner function |
|---|---|---|
| \((3x^2+1)^5\) | \((\square)^5\) | \(3x^2+1\) |
| \(\sin(4x)\) | \(\sin(\square)\) | \(4x\) |
| \(\cos(x^3)\) | \(\cos(\square)\) | \(x^3\) |
| \(e^{2x}\) | \(e^{\square}\) | \(2x\) |
| \(\ln(5x+1)\) | \(\ln(\square)\) | \(5x+1\) |
Two patterns are worth naming explicitly.
Power composites. When a whole expression is raised to a power, the power is the outer function. This includes disguised powers: \(\sqrt{x^2+1}\) is \((x^2+1)^{1/2}\), so the outer function is the one-half power and the inner function is \(x^2+1\).
Trig, exponential, and log composites. When a trig, exponential, or log function is applied to something other than plain \(x\), that function is the outer function. The test is simple: is the argument of the function just \(x\)? If not, you have a composite.
A useful warning: \(\sin x \cdot x^2\) is not a composite at all. It is a product of two separate functions, and the outer/inner language does not apply. Composition means one function is inside another, not multiplied by it.