To find the given derivative [tex]\(\left.\frac{d}{d x}[x f(x)]\right|_{x=4}\)[/tex] using the information from the table, we will use the product rule for differentiation.
The product rule states that if you have two functions [tex]\(u(x)\)[/tex] and [tex]\(v(x)\)[/tex], the derivative of their product [tex]\(u(x) v(x)\)[/tex] is given by:
[tex]\[
\frac{d}{dx}[u(x) v(x)] = u(x) \cdot \frac{d}{dx}[v(x)] + v(x) \cdot \frac{d}{dx}[u(x)]
\][/tex]
In our case, [tex]\(u(x) = x\)[/tex] and [tex]\(v(x) = f(x)\)[/tex]. Applying the product rule:
[tex]\[
\frac{d}{dx}[x f(x)] = x \cdot \frac{d}{dx}[f(x)] + f(x) \cdot \frac{d}{dx}[x]
\][/tex]
We know that [tex]\(\frac{d}{dx}[x] = 1\)[/tex], so this simplifies to:
[tex]\[
\frac{d}{dx}[x f(x)] = x \cdot f'(x) + f(x)
\][/tex]
Now we need to evaluate this expression at [tex]\(x = 4\)[/tex]. From the table:
[tex]\[
f(4) = 3 \quad \text{and} \quad f'(4) = 5
\][/tex]
Substituting [tex]\(x = 4\)[/tex], [tex]\(f(4)\)[/tex], and [tex]\(f'(4)\)[/tex] into the derivative expression, we get:
[tex]\[
\left. \frac{d}{dx}[x f(x)] \right|_{x=4} = 4 \cdot 5 + 3
\][/tex]
Simplifying this:
[tex]\[
4 \cdot 5 = 20
\][/tex]
[tex]\[
20 + 3 = 23
\][/tex]
So, the value of the derivative at [tex]\(x = 4\)[/tex] is:
[tex]\[
\left.\frac{d}{d x}[x f(x)]\right|_{x=4}= 23
\][/tex]
