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We are asked to find the derivative of the function \( f(x) = \sqrt{x} \cdot e^{6x} \).

### Step-by-Step Solution:

The function can be rewritten as:
\[
f(x) = x^{1/2} \cdot e^{6x}
\]

We'll apply the **product rule** for differentiation, which states:
\[
(f \cdot g)' = f' \cdot g + f \cdot g'
\]
Here, \( f(x) = x^{1/2} \) and \( g(x) = e^{6x} \).

#### Step 1: Differentiate \( x^{1/2} \)
The derivative of \( x^{1/2} \) is:
\[
\frac{d}{dx} \left( x^{1/2} \right) = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}}
\]

#### Step 2: Differentiate \( e^{6x} \)
The derivative of \( e^{6x} \) is:
\[
\frac{d}{dx} \left( e^{6x} \right) = 6 e^{6x}
\]

#### Step 3: Apply the product rule
Now, apply the product rule:
\[
f'(x) = \left( \frac{1}{2\sqrt{x}} \cdot e^{6x} \right) + \left( x^{1/2} \cdot 6 e^{6x} \right)
\]
Simplifying both terms:

\[
f'(x) = \frac{e^{6x}}{2\sqrt{x}} + 6\sqrt{x} e^{6x}
\]

#### Step 4: Combine terms
Factor out \( e^{6x} \):
\[
f'(x) = e^{6x} \left( \frac{1}{2\sqrt{x}} + 6\sqrt{x} \right)
\]

Now, express everything under a common denominator:
\[
f'(x) = e^{6x} \left( \frac{1 + 12x}{2\sqrt{x}} \right)
\]

This matches the option:
\[
\boxed{(6 \cdot x + 1) \cdot e^{6x} / 2\sqrt{x}}
\]

This corresponds to the second option in the image.

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Would you like further clarification or any additional details on this problem?

Here are some related questions to explore:
1. What is the product rule in calculus, and when do we use it?
2. How do we differentiate exponential functions involving constants?
3. What is the chain rule, and how would it apply if the exponent in \( e^{6x} \) were more complex?
4. How can we simplify expressions involving radicals and exponents for derivatives?
5. Why is factoring out common terms, like \( e^{6x} \), helpful in derivative expressions?

**Tip**: When applying the product rule, always simplify intermediate results as much as possible before moving on to the next step!

What is the derivative of f(x) = √x * e^(6x)?

This page explains how to differentiate the function f(x) = √x * e^(6x) using the product rule. The step-by-step solution includes applying the product rule, simplifying the result, and factoring out common terms. Ideal for advanced high school or early university students studying calculus.

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