For the given function:

$f(x) = 2e^{4x+2}$

We are tasked to find the derivative $f'(x)$.

Solution:

The function involves an exponential term, so we apply the chain rule. The derivative of $e^{u}$ is $e^u \cdot u'$, where $u = 4x+2$.

1. Rewrite the function: $f(x) = 2e^{4x+2}$

2. Compute the derivative of $4x+2$: $u = 4x + 2 \quad \Rightarrow \quad u' = 4$

3. Apply the chain rule: $f'(x) = 2 \cdot e^{4x+2} \cdot 4$

4. Simplify the expression: $f'(x) = 8e^{4x+2}$

Final Answer:

$f'(x) = 8e^{4x+2}$

Let me know if you need further details or explanations! Here are related questions to explore:

1. How do you differentiate exponential functions with more complex exponents?
2. How does the chain rule apply to functions of multiple variables?
3. Can you integrate $f(x) = 2e^{4x+2}$?
4. What is the behavior of $f'(x)$ as $x \to \infty$?
5. How does the derivative $f'(x)$ relate to the slope of $f(x)$?

Tip: Always simplify exponents before differentiation to make the process more straightforward.