The problem asks to compute the derivative of the difference between two functions, $f(x)$ and $g(x)$, evaluated at $x = 4$, using the given table.

We are tasked with finding: $\frac{d}{dx} [f(x) - g(x)] \Big|_{x=4}$

Step-by-step Solution:

1. Derivative of $f(x) - g(x)$: The derivative of the difference of two functions is: $\frac{d}{dx} [f(x) - g(x)] = f'(x) - g'(x)$

2. Evaluate at $x = 4$: According to the table:

   • $f'(4) = 3$
   • $g'(4) = 2$

   Thus, we compute: $f'(4) - g'(4) = 3 - 2 = 1$

Therefore, the value of the derivative at $x = 4$ is $1$.

Would you like any further details or explanations?

Relative questions:

1. What is the general rule for differentiating the sum or difference of two functions?
2. How do you apply the chain rule in more complex cases?
3. Why does subtracting two functions not affect the differentiation process?
4. How can you use a table like this for more advanced derivative operations?
5. What other methods could you use to check or verify this derivative?

Tip:

When differentiating combinations of functions, always break them down into smaller parts using basic rules like the sum, difference, and product rules.