https://r2.erweima.ai/mathbot/image/17260977780659fnmecti.jpg

Let's solve each of the questions (8 to 10) provided in the image one by one using the given table and functions.

### Question 8
Given:
- \( f(x) = x^2 + 3x \)
- \( g(x) \) and its derivative \( g'(x) \) have values in the table.
- \( h(x) = \frac{f(x)}{g(x)} \)
  
We need to find \( h'(2) \).

To find \( h'(x) \), we use the quotient rule:

\[
h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}
\]

First, calculate \( f'(x) \):
\[
f'(x) = \frac{d}{dx} (x^2 + 3x) = 2x + 3
\]

Now, substitute \( x = 2 \):

- \( f(2) = 2^2 + 3 \cdot 2 = 4 + 6 = 10 \)
- \( f'(2) = 2 \cdot 2 + 3 = 4 + 3 = 7 \)
- \( g(2) = 1 \) (from the table)
- \( g'(2) = 3 \) (from the table)

Substitute these values into the derivative formula:

\[
h'(2) = \frac{f'(2)g(2) - f(2)g'(2)}{(g(2))^2} = \frac{7 \cdot 1 - 10 \cdot 3}{(1)^2} = \frac{7 - 30}{1} = -23
\]

So, the answer is **A. -23**.

### Question 9
Given:
- \( f(x) = \sqrt{x} \)
- \( H(x) = f(x)g(x) \)

We need to find \( H'(4) \).

To find \( H'(x) \), we use the product rule:

\[
H'(x) = f'(x)g(x) + f(x)g'(x)
\]

First, find \( f'(x) \):

\[
f'(x) = \frac{d}{dx} (\sqrt{x}) = \frac{1}{2\sqrt{x}}
\]

Now, substitute \( x = 4 \):

- \( f(4) = \sqrt{4} = 2 \)
- \( f'(4) = \frac{1}{2\sqrt{4}} = \frac{1}{4} \)
- \( g(4) = 7 \) (from the table)
- \( g'(4) = -1 \) (from the table)

Substitute these values into the derivative formula:

\[
H'(4) = f'(4)g(4) + f(4)g'(4) = \frac{1}{4} \cdot 7 + 2 \cdot (-1) = \frac{7}{4} - 2 = \frac{7}{4} - \frac{8}{4} = -\frac{1}{4}
\]

So, the answer is **C. -1/4**.

### Question 10
Given:
- \( g(4) = 7 \)
- We need the tangent line approximation for \( g(3.9) \).

The tangent line approximation at \( x = 4 \) for \( g(x) \) is given by:

\[
g(x) \approx g(4) + g'(4)(x - 4)
\]

Substitute the values from the table:

- \( g(4) = 7 \)
- \( g'(4) = -1 \)

Now, find the approximation for \( g(3.9) \):

\[
g(3.9) \approx 7 + (-1)(3.9 - 4) = 7 + (-1)(-0.1) = 7 + 0.1 = 7.1
\]

So, the answer is **B. 7.1**.

### Summary of Answers:
- **8. A. -23**
- **9. C. -1/4**
- **10. B. 7.1**

Would you like any further details or explanations on any of the questions? Here are 5 related questions to deepen your understanding:

1. How is the quotient rule used to differentiate a function?
2. What is the product rule for differentiation and how does it apply to functions like \( H(x) = f(x)g(x) \)?
3. How do you use the tangent line approximation to estimate values of a function near a known point?
4. Can you find the derivative of any other functions using the product or quotient rule?
5. How can you determine the concavity of a function using its second derivative?

**Tip:** Remember, the quotient rule is particularly useful when differentiating ratios of functions, and the product rule is great for products of functions. Practice these rules to master their application in different scenarios.

Use the table below to answer questions 8-10 regarding the functions f(x), g(x), and H(x).

Solve calculus problems involving the quotient rule, product rule, and tangent line approximation using functions and their derivatives. Practice key concepts in calculus with step-by-step solutions for differentiation and approximation.

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