To find [tex]\( h'(x) \)[/tex] where [tex]\( h(x) = \frac{f(x)}{g(x)} \)[/tex], we can use the quotient rule for differentiation. The quotient rule states that if [tex]\( h(x) = \frac{f(x)}{g(x)} \)[/tex], then
[tex]\[
h'(x) = \frac{f'(x) g(x) - f(x) g'(x)}{(g(x))^2}
\][/tex]
We are given the values at [tex]\( x = 2 \)[/tex] from the table:
[tex]\[
f(2) = -4, \quad g(2) = 6, \quad f'(2) = 3, \quad g'(2) = 7
\][/tex]
Substitute these values into the quotient rule formula:
1. Calculate the numerator:
[tex]\[
\text{Numerator} = f'(2) \cdot g(2) - f(2) \cdot g'(2)
\][/tex]
[tex]\[
\text{Numerator} = 3 \cdot 6 - (-4) \cdot 7
\][/tex]
[tex]\[
\text{Numerator} = 18 + 28
\][/tex]
[tex]\[
\text{Numerator} = 46
\][/tex]
2. Calculate the denominator:
[tex]\[
\text{Denominator} = (g(2))^2
\][/tex]
[tex]\[
\text{Denominator} = 6^2
\][/tex]
[tex]\[
\text{Denominator} = 36
\][/tex]
3. Divide the numerator by the denominator:
[tex]\[
h'(2) = \frac{46}{36}
\][/tex]
4. Simplify and round the result to 3 decimal places:
[tex]\[
h'(2) \approx 1.278
\][/tex]
Thus, the derivative [tex]\( h'(2) \)[/tex] is approximately [tex]\( 1.278 \)[/tex].
