To find [tex]\((f \cdot g)^{\prime}(2)\)[/tex], we need to use the product rule for derivatives. The product rule states:
[tex]\[(f \cdot g)^{\prime}(x) = f^{\prime}(x) \cdot g(x) + f(x) \cdot g^{\prime}(x)\][/tex]
Given the table with the following values at [tex]\(x = 2\)[/tex]:
[tex]\[ f(2) = 4 \][/tex]
[tex]\[ f^{\prime}(2) = -6 \][/tex]
[tex]\[ g(2) = -10 \][/tex]
[tex]\[ g^{\prime}(2) = 1 \][/tex]
We can substitute these values into the product rule formula.
Step-by-step calculation:
1. Calculate [tex]\(f^{\prime}(2) \cdot g(2)\)[/tex]:
[tex]\[ f^{\prime}(2) \cdot g(2) = -6 \cdot (-10) = 60 \][/tex]
2. Calculate [tex]\(f(2) \cdot g^{\prime}(2)\)[/tex]:
[tex]\[ f(2) \cdot g^{\prime}(2) = 4 \cdot 1 = 4 \][/tex]
3. Add the results from steps 1 and 2:
[tex]\[ (f \cdot g)^{\prime}(2) = 60 + 4 = 64 \][/tex]
So, [tex]\((f \cdot g)^{\prime}(2) = 64\)[/tex].
