To find [tex]\( h'(1) \)[/tex] where [tex]\( h(x) = f(x) \cdot g(x) \)[/tex], we need to use the product rule for differentiation. The product rule states:
[tex]\[
h'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)
\][/tex]
Given the values from the table for [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = -5
\][/tex]
[tex]\[
g(1) = 4
\][/tex]
[tex]\[
f'(1) = 2
\][/tex]
[tex]\[
g'(1) = 6
\][/tex]
Now, we apply these values to the product rule formula:
[tex]\[
h'(1) = f'(1) \cdot g(1) + f(1) \cdot g'(1)
\][/tex]
Substitute the values into the formula:
[tex]\[
h'(1) = 2 \cdot 4 + (-5) \cdot 6
\][/tex]
Calculate each term:
[tex]\[
2 \cdot 4 = 8
\][/tex]
[tex]\[
-5 \cdot 6 = -30
\][/tex]
Add the results:
[tex]\[
h'(1) = 8 + (-30) = 8 - 30 = -22
\][/tex]
Therefore, the value of [tex]\( h'(1) \)[/tex] is:
[tex]\[
h'(1) = -22
\][/tex]
So, the detailed solution shows that [tex]\( h'(1) = -22 \)[/tex].
