To solve problem A, which asks for the derivative of the following function:

$y = \frac{e^{4x}}{2}$

we'll apply the following steps:

Step 1: Constant Multiple Rule

The function can be written as: $y = \frac{1}{2} e^{4x}$ This separates the constant $\frac{1}{2}$ from the exponential function $e^{4x}$.

When taking derivatives, we can factor out constants. So, we focus on differentiating $e^{4x}$.

Step 2: Derivative of $e^{4x}$

The derivative of an exponential function $e^{g(x)}$ is given by: $\frac{d}{dx}[e^{g(x)}] = e^{g(x)} \cdot g'(x)$ Here, $g(x) = 4x$, so the derivative of $g(x)$ is: $g'(x) = 4$

Thus, the derivative of $e^{4x}$ is: $\frac{d}{dx}[e^{4x}] = e^{4x} \cdot 4 = 4e^{4x}$

Step 3: Multiply by the Constant

Now, multiply this result by the constant $\frac{1}{2}$ that we factored out earlier: $\frac{d}{dx}\left[\frac{1}{2} e^{4x}\right] = \frac{1}{2} \cdot 4e^{4x} = 2e^{4x}$

Final Answer:

Thus, the derivative of $y = \frac{e^{4x}}{2}$ is: $y' = 2e^{4x}$

Would you like more details on this? Here are some follow-up questions:

1. What is the general rule for differentiating exponential functions like $e^{ax}$?
2. How would the answer change if the exponent were $4x^2$ instead of $4x$?
3. What happens if the constant factor in front is not $1/2$ but some other value, say $3$?
4. How do we handle cases where the exponent is a more complex function of $x$?
5. What is the derivative of $e^{-4x}$?

Tip: When differentiating exponential functions of the form $e^{g(x)}$, always remember to use the chain rule by multiplying by the derivative of $g(x)$.