We are tasked with finding the derivative of the function $f(x) = e^{x \sqrt{3x + 2}}$. Let's go step by step.

Function:

$f(x) = e^{x \sqrt{3x + 2}}$

Step 1: Use the chain rule

Since the function is in the form $f(x) = e^{g(x)}$, where $g(x) = x \sqrt{3x + 2}$, the derivative of $e^{g(x)}$ is:

$f'(x) = e^{g(x)} \cdot g'(x)$

So, we need to find $g'(x)$, where $g(x) = x \sqrt{3x + 2}$.

Step 2: Apply product and chain rule to $g(x)$

We can see that $g(x) = x \sqrt{3x + 2}$ is a product of two functions, $x$ and $\sqrt{3x + 2}$. Use the product rule:

$g'(x) = \frac{d}{dx} \left( x \right) \cdot \sqrt{3x + 2} + x \cdot \frac{d}{dx} \left( \sqrt{3x + 2} \right)$

Step 3: Differentiate $\sqrt{3x + 2}$

Now, differentiate $\sqrt{3x + 2}$. Using the chain rule:

$\frac{d}{dx} \left( \sqrt{3x + 2} \right) = \frac{1}{2\sqrt{3x + 2}} \cdot 3 = \frac{3}{2\sqrt{3x + 2}}$

Step 4: Apply product rule

Now substitute back into the product rule:

$g'(x) = 1 \cdot \sqrt{3x + 2} + x \cdot \frac{3}{2\sqrt{3x + 2}} = \sqrt{3x + 2} + \frac{3x}{2\sqrt{3x + 2}}$

Simplify by getting a common denominator:

$g'(x) = \frac{2(3x + 2) + 3x}{2\sqrt{3x + 2}} = \frac{6x + 4 + 3x}{2\sqrt{3x + 2}} = \frac{9x + 4}{2\sqrt{3x + 2}}$

Step 5: Compute $f'(x)$

Now, substitute $g'(x)$ into the derivative formula for $f(x)$:

$f'(x) = e^{x \sqrt{3x + 2}} \cdot \frac{9x + 4}{2\sqrt{3x + 2}}$

Thus, the derivative of the function is:

$f'(x) = e^{x \sqrt{3x + 2}} \cdot \frac{9x + 4}{2\sqrt{3x + 2}}$

Would you like further details on any part of the process?

Here are 5 follow-up questions related to this problem:

1. What is the chain rule and how is it applied in this case?
2. Can you explain how the product rule works in differentiating $g(x)$?
3. Why do we need to simplify the derivative of $\sqrt{3x + 2}$?
4. What is the significance of exponential functions when differentiating?
5. How would the derivative change if the exponent were a different function?

Tip: When differentiating products of functions, always check if either the product rule or chain rule applies to simplify your steps.