Let's go through the problems shown in the image one by one.

Problem 20:

You are asked to find $f'(5)$ if $f(x) = \frac{h(x)}{2g(x)}$. To solve this, we will use the quotient rule for differentiation. The quotient rule states:

$\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}$

Here:

• $u(x) = h(x)$
• $v(x) = 2g(x)$

Applying the quotient rule:

$f'(x) = \frac{h'(x) \cdot 2g(x) - h(x) \cdot 2g'(x)}{(2g(x))^2}$

Now substitute the values given in the table for $x = 5$:

• $h(5) = 3$
• $h'(5) = -\frac{1}{2}$
• $g(5) = 2$
• $g'(5) = 8$

Substitute these into the formula:

$f'(5) = \frac{-\frac{1}{2} \cdot 2 \cdot 2 - 3 \cdot 2 \cdot 8}{(2 \cdot 2)^2}$

Simplify:

$f'(5) = \frac{-2 - 48}{16} = \frac{-50}{16} = -\frac{25}{8}$

Thus, the derivative $f'(5) = -\frac{25}{8}$.

Would you like a detailed explanation for other problems, such as number 17, 18, or 19? Here are 5 relative questions that you may want to explore next:

1. How do you solve problem 19, which asks for the tangent line equation?
2. Can we break down the function in problem 17 for differentiation?
3. How can you solve for $f'(x)$ in problem 21 using the product rule?
4. Why is the quotient rule essential in problems involving rational functions like problem 20?
5. What are some common mistakes students make when using the chain or quotient rule?

Tip: When applying the quotient rule, remember to keep the order of the numerator's terms consistent to avoid sign errors.