The question involves finding the derivative $\frac{d}{dx} \left( \frac{f(x)}{x + 5} \right)$ at $x = 1$ using the values from a table.

Steps to Solve the Problem

To find $\frac{d}{dx} \left( \frac{f(x)}{x + 5} \right)$, we can apply the Quotient Rule for derivatives, which states:

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

Here:

• $u(x) = f(x)$ and $u'(x) = f'(x)$.
• $v(x) = x + 5$ and $v'(x) = 1$.

Step 1: Apply the Quotient Rule

Using the Quotient Rule, we get:

$\frac{d}{dx} \left( \frac{f(x)}{x + 5} \right) = \frac{f'(x) (x + 5) - f(x) \cdot 1}{(x + 5)^2}$

Step 2: Substitute $x = 1$

Substitute $x = 1$ and the values of $f(1)$, $f'(1)$ from the table:

• $f(1) = 5$
• $f'(1) = 3$

Now, substitute these values into the derivative:

$\frac{d}{dx} \left( \frac{f(x)}{x + 5} \right) \bigg|_{x=1} = \frac{3 \cdot (1 + 5) - 5 \cdot 1}{(1 + 5)^2}$

Step 3: Simplify the Expression

1. Compute the numerator: $3 \cdot 6 - 5 = 18 - 5 = 13$
2. Compute the denominator: $(1 + 5)^2 = 6^2 = 36$

Thus, the derivative at $x = 1$ is: $\frac{13}{36}$

So, the answer is $\frac{13}{36}$.

Would you like more details or have any questions about the solution?

Related Questions:

1. How do you apply the Quotient Rule to different types of functions?
2. What are some common mistakes to avoid when using the Quotient Rule?
3. How can you find the derivative of a product of two functions?
4. What is the Chain Rule, and how does it differ from the Quotient Rule?
5. How would you approach finding the second derivative of $\frac{f(x)}{x + 5}$?

Tip:

When using the Quotient Rule, remember to always square the denominator and carefully apply the derivatives to both the numerator and the denominator.