d/dx{ln e^{2x}} = \na 1/e^{2x}\nb 2/e^{2x}\nc 2x\nd 1\ne 2

Explanation:

Step1: Simplify the inner - function

Use the property $\ln(e^a)=a$. So, $\ln(e^{2x}) = 2x$.

Step2: Differentiate the simplified function

The derivative of $y = 2x$ with respect to $x$ is $\frac{d}{dx}(2x)=2$ (using the power - rule $\frac{d}{dx}(ax)=a$ where $a = 2$).

Answer:

E. 2