Step 1: Understanding the Concept:
This problem requires finding the derivative of a composite exponential function where the power component is itself a distinct function of $x$. To correctly differentiate a composite function of the form $f(g(x))$, we must use the Chain Rule from differential calculus.
Step 2: Key Formula or Approach:
- The derivative of the baseline natural exponential function $e^u$ with respect to its variable $u$ is itself:
$$ \frac{d}{du}(e^u) = e^u $$
- The Chain Rule dictates that if a variable $y$ depends on $u$, which in turn depends on $x$, then:
$$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} $$
Consequently, the derivative rule for an exponential function containing a linear function in its power is $\frac{d}{dx}(e^{ax}) = a e^{ax}$.
Step 3: Detailed Explanation:
We are given the exponential function:
$$ y = e^{2x} $$
Let's define the inner exponent function as a temporary variable $u$:
$$ u = 2x $$
This substitution allows us to express our main function as $y = e^u$. Now we differentiate both components separately:
1. Differentiate the outer function $y$ with respect to our inner variable $u$:
$$ \frac{dy}{du} = \frac{d}{du}(e^u) = e^u $$
2. Differentiate our inner function $u$ with respect to the primary variable $x$:
$$ \frac{du}{dx} = \frac{d}{dx}(2x) = 2 $$
Finally, apply the Chain Rule formula by computing the product of these two derivatives:
$$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = e^u \cdot 2 = 2e^u $$
Substituting our original expression for $u$ ($u = 2x$) back into the equation yields:
$$ \frac{dy}{dx} = 2e^{2x} $$
This result corresponds precisely to choice (A).
Step 4: Final Answer:
The derivative $\frac{dy}{dx}$ is equal to $2e^{2x}$.