To find $h'(3)$ where $h(x) = \frac{f(x)}{g(x)}$, we use the quotient rule for differentiation, which states:

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

Given values:

• $f(3) = 2$
• $f'(3) = -1$
• $g(3) = 4$
• $g'(3) = 5$

Applying these values to $h'(3)$:

$h'(3) = \frac{f'(3) \cdot g(3) - f(3) \cdot g'(3)}{[g(3)]^2}$

Substitute the values:

$h'(3) = \frac{(-1) \cdot 4 - 2 \cdot 5}{4^2}$

Calculate each term:

1. $(-1) \cdot 4 = -4$
2. $2 \cdot 5 = 10$
3. $4^2 = 16$

Now, substitute these into the expression:

$h'(3) = \frac{-4 - 10}{16} = \frac{-14}{16} = -\frac{7}{8}$

Final Answer:

$h'(3) = -\frac{7}{8}$

Would you like more details or have any questions?

1. What is the quotient rule, and how does it apply to similar problems?
2. How does the chain rule differ from the quotient rule in calculus?
3. Why is it important to know both $f(x)$ and $g(x)$ values when differentiating a quotient?
4. How would the answer change if $g(3)$ was zero?
5. What are other ways to differentiate functions that involve division?

Tip: When working with derivatives, always double-check for correct substitution, especially with negative values, to avoid calculation errors.