The layout on exponential and logarithmic composites
Example 1: \(e^{2x}\)
- Outer layer: \(e^{\square}\), whose derivative is \(e^{\square}\) — the exponential reproduces itself.
- Inner layer: \(2x\), whose derivative is \(2\).
\[\frac{d}{dx}e^{2x} = e^{2x} \cdot 2\]
Because the outer derivative is the exponential itself, the outer factor looks identical to the original function. That is not a mistake; it is the defining property of \(e^{\square}\). The inner factor \(2\) is what actually changes the answer.
Example 2: \(\ln(5x+1)\)
- Outer layer: \(\ln(\square)\), whose derivative is \(\dfrac{1}{\square}\).
- Inner layer: \(5x+1\), whose derivative is \(5\).
\[\frac{d}{dx}\ln(5x+1) = \frac{1}{5x+1} \cdot 5\]
The outer factor is \(\dfrac{1}{5x+1}\), not \(\dfrac{1}{x}\). The denominator must be the whole inner expression. This is the same trap that \(\ln(x^2)\) sets: writing \(\dfrac{1}{x}\) instead of \(\dfrac{1}{x^2}\) evaluates the outer derivative at the wrong place.
The pattern
| Composite | Outer factor | Inner factor |
|---|---|---|
| \(e^{2x}\) | \(e^{2x}\) | \(2\) |
| \(\ln(5x+1)\) | \(\dfrac{1}{5x+1}\) | \(5\) |
Exponential and logarithmic composites differ from power and trig composites only in what the outer derivative looks like. The slot structure is unchanged.