To differentiate the function $y = e^{4x}$, we will apply the chain rule.

The chain rule states that if you have a composition of functions $f(g(x))$, then the derivative is given by: $\frac{d}{dx} \left( f(g(x)) \right) = f'(g(x)) \cdot g'(x)$

For $y = e^{4x}$, we can think of this as $f(u) = e^u$ where $u = 4x$. So, the derivative of $e^{4x}$ is:

1. The derivative of $e^u$ is $e^u$ itself.
2. The derivative of $4x$ with respect to $x$ is 4.

Thus, applying the chain rule:

$\frac{dy}{dx} = e^{4x} \cdot 4 = 4e^{4x}$

So, the derivative of $y = e^{4x}$ is:

$\frac{dy}{dx} = 4e^{4x}$

Let me know if you'd like further explanation or clarification! Here are a few related questions:

1. How do you differentiate $y = e^{ax}$, where $a$ is a constant?
2. What is the derivative of $y = e^{x^2}$?
3. How do you apply the chain rule to more complex functions?
4. What is the derivative of $y = e^{x^2 + 3x}$?
5. How do you use the product rule with exponential functions?

Tip: When differentiating exponential functions, always remember that the exponential function $e^x$ is its own derivative, and you apply the chain rule for any function in the exponent.