https://r2.erweima.ai/mathbot/image/1727221828860l8twzzte.jpg

The function given in the image is:

\[
g(s) = \sqrt{s}(s^2 + 8)
\]

To find the derivative of this function using the product rule, we will treat it as a product of two functions:

1. \( f(s) = \sqrt{s} = s^{1/2} \)
2. \( h(s) = s^2 + 8 \)

### Step-by-Step Derivative Using the Product Rule

The product rule states:

\[
\frac{d}{ds}[f(s) \cdot h(s)] = f'(s) \cdot h(s) + f(s) \cdot h'(s)
\]

#### Step 1: Find \( f'(s) \)
The derivative of \( f(s) = s^{1/2} \) is:

\[
f'(s) = \frac{1}{2}s^{-1/2}
\]

#### Step 2: Find \( h'(s) \)
The derivative of \( h(s) = s^2 + 8 \) is:

\[
h'(s) = 2s
\]

#### Step 3: Apply the Product Rule
Now, applying the product rule:

\[
g'(s) = f'(s) \cdot h(s) + f(s) \cdot h'(s)
\]

Substitute the values:

\[
g'(s) = \left(\frac{1}{2}s^{-1/2}\right) \cdot (s^2 + 8) + \left(s^{1/2}\right) \cdot (2s)
\]

#### Step 4: Simplify the Expression
Simplify each term:

\[
g'(s) = \frac{1}{2}s^{-1/2}(s^2 + 8) + 2s^{3/2}
\]

Expanding the terms:

\[
g'(s) = \frac{1}{2}s^{-1/2}(s^2) + \frac{1}{2}s^{-1/2}(8) + 2s^{3/2}
\]

\[
g'(s) = \frac{1}{2}s^{3/2} + 4s^{-1/2} + 2s^{3/2}
\]

Finally, combine like terms:

\[
g'(s) = \left(\frac{1}{2} + 2\right)s^{3/2} + 4s^{-1/2}
\]

\[
g'(s) = \frac{5}{2}s^{3/2} + 4s^{-1/2}
\]

This is the derivative of the function \( g(s) \).

---

Do you need further clarification or more details? Here are 5 related questions for practice:

1. What is the product rule for derivatives?
2. How do you differentiate powers of \( s \), such as \( s^{1/2} \)?
3. Can you apply the product rule to a function with more than two terms?
4. What are the steps to simplify a derivative involving fractional exponents?
5. How do you interpret the derivative in the context of a real-world problem?

**Tip:** When applying the product rule, always make sure to clearly identify each part of the product before taking derivatives.

Usar la regla del producto para calcular la derivada de la función g(s) = √s(s² + 8).

Learn how to apply the product rule to differentiate the function g(s) = √s(s² + 8). This detailed solution covers key calculus concepts, including the product rule and simplifying derivatives with fractional exponents. Suitable for high school students in Grades 11-12.

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