The layout on a power composite
A power of a function, such as \((3x^2+1)^5\), is a composite with two layers:
- Outer function: the fifth power, \((\square)^5\).
- Inner function: the polynomial \(3x^2+1\).
Step 1 — Write the outer factor
Differentiate the outer function while copying the inner expression in unchanged:
\[5(3x^2+1)^4\]
The exponent drops by one, and the base stays exactly \(3x^2+1\). It does not become \(3x^2\) or \(x^2+1\).
Step 2 — Write the inner factor
Differentiate the inner function as its own separate factor:
\[\frac{d}{dx}(3x^2+1) = 6x\]
Step 3 — Multiply, then stop
The complete unsimplified derivative is
\[\frac{d}{dx}(3x^2+1)^5 = 5(3x^2+1)^4 \cdot 6x\]
Two layers, two factors. Simplifying to \(30x(3x^2+1)^4\) is optional and can wait until the factor count has been checked.
Why the layout matters here
The most common error on a power composite is differentiating the outer power and forgetting the inner factor entirely, giving \(5(3x^2+1)^4\) and stopping. Writing the inner factor as a separate slot makes that omission visible: the second slot is empty.