We are given the following information:

• $f(1) = 5$
• $f'(1) = 9$
• $g(1) = -1$
• $g'(1) = 8$

We are tasked with finding the value of $\left( \frac{f}{g} \right)'(1)$.

Step-by-step Solution:

To solve this, we will use the quotient rule for derivatives. The quotient rule states that for two functions $f(x)$ and $g(x)$, the derivative of their quotient is:

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

Now, applying this rule at $x = 1$:

$\left( \frac{f}{g} \right)'(1) = \frac{f'(1) g(1) - f(1) g'(1)}{g(1)^2}$

Plug in the given values:

• $f'(1) = 9$
• $g(1) = -1$
• $f(1) = 5$
• $g'(1) = 8$

Now substitute these into the quotient rule formula:

$\left( \frac{f}{g} \right)'(1) = \frac{(9)(-1) - (5)(8)}{(-1)^2}$

Simplify the numerator:

= \frac{-49}{1} = -49$$ ### Final Answer: $$\left( \frac{f}{g} \right)'(1) = -49$$ --- Would you like a more detailed explanation or have any further questions on this? Here's some follow-up content: 1. What is the quotient rule in general? 2. Can the quotient rule be applied when $$g(x) = 0$$? 3. How does the product rule differ from the quotient rule? 4. What happens to the derivative when $$f(x)$$ or $$g(x)$$ is a constant? 5. How would the quotient rule apply in cases of higher-order derivatives? **Tip**: Always double-check if $$g(x)$$ is nonzero before applying the quotient rule, as division by zero is undefined.