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Find derivatives of the functions defined as follows.  Show step by step 

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.

Find the derivative of the function f(x) = e^{x \sqrt{3x + 2}}.

Learn how to find the derivative of the function f(x) = e^{x√(3x+2)} step by step, using calculus principles like the chain rule and product rule. This problem is suitable for advanced high school or undergraduate calculus students.

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