Question

$\frac{d}{dx}(\ln e^{2x}) = 2$

Explanation:

Step1: Simplify the logarithm

Using the property of logarithms $\ln(e^u)=u$, we simplify $\ln(e^{2x})$ to $2x$. So the function becomes $\frac{d}{dx}(2x)$.

Step2: Differentiate the simplified function

The derivative of $2x$ with respect to $x$ is given by the power rule $\frac{d}{dx}(ax^n)=nax^{n - 1}$. Here, $a = 2$ and $n=1$, so $\frac{d}{dx}(2x)=2\times1\times x^{1 - 1}=2$.

Answer:

The derivative $\frac{d}{dx}(\ln e^{2x})$ is indeed equal to 2, so the statement is true.