To find [tex]\(\left(\frac{f}{g}\right)^{\prime}(5)\)[/tex], we need to apply the quotient rule for differentiation. The quotient rule states that if you have a function [tex]\(h(x) = \frac{f(x)}{g(x)}\)[/tex], then its derivative [tex]\(h'(x)\)[/tex] can be found using the formula:
[tex]\[
\left(\frac{f}{g}\right)'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}
\][/tex]
Given the values from the table at [tex]\(x = 5\)[/tex]:
- [tex]\(f(5) = 3\)[/tex]
- [tex]\(f'(5) = 6\)[/tex]
- [tex]\(g(5) = 4\)[/tex]
- [tex]\(g'(5) = -7\)[/tex]
We substitute these values into the quotient rule formula as follows:
1. Calculate the numerator [tex]\(f'(x)g(x) - f(x)g'(x)\)[/tex]:
[tex]\[
f'(5)g(5) - f(5)g'(5) = 6 \cdot 4 - 3 \cdot (-7)
\][/tex]
[tex]\[
= 24 + 21
\][/tex]
[tex]\[
= 45
\][/tex]
2. Calculate the denominator [tex]\(g(x)^2\)[/tex]:
[tex]\[
g(5)^2 = 4^2 = 16
\][/tex]
3. Divide the numerator by the denominator to find the derivative:
[tex]\[
\left(\frac{f}{g}\right)'(5) = \frac{45}{16}
\][/tex]
[tex]\[
= 2.8125
\][/tex]
Therefore, the derivative [tex]\(\left(\frac{f}{g}\right)'(5)\)[/tex] is [tex]\(2.8125\)[/tex]. If rounded to two decimal places, the final answer is:
[tex]\[
\left(\frac{f}{g}\right)'(5) = 2.81
\][/tex]
