We are given the functions $f(x)$ and $g(x)$ with certain values at $x = 4$, and we're tasked with finding $h'(4)$ for $h(x) = \frac{f(x)}{g(x)}$.

Step 1: Use the Quotient Rule for Derivatives

The derivative of $h(x) = \frac{f(x)}{g(x)}$ is given by the quotient rule:

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

Step 2: Plug in the values at $x = 4$

We are given:

• $f(4) = 2$
• $g(4) = 5$
• $f'(4) = 6$
• $g'(4) = -3$

Substitute these values into the quotient rule formula:

$h'(4) = \frac{f'(4)g(4) - f(4)g'(4)}{(g(4))^2}$

$h'(4) = \frac{6 \times 5 - 2 \times (-3)}{5^2}$

Step 3: Simplify the expression

$h'(4) = \frac{30 + 6}{25} = \frac{36}{25}$

This simplifies to:

$h'(4) = 1.44$

Final Answer:

The value of $h'(4)$ is $1.44$ (exact decimal value).

Would you like any additional details or clarification?

Here are 5 related questions to deepen your understanding:

1. What is the quotient rule and when is it used?
2. How do you apply the product rule in differentiation?
3. Can the quotient rule be used for three functions?
4. What is the geometric interpretation of $h'(4)$?
5. How would the solution change if $g(4)$ were 0?

Tip: Always verify the conditions (such as non-zero denominator) before applying the quotient rule to avoid undefined expressions.