Final answer:
To find (fg)'(2), use the product rule of differentiation to evaluate f'(2) multiplied by g(2), plus f(2) multiplied by g'(2). To find (f/g)'(2), use the quotient rule of differentiation to evaluate f'(2) multiplied by g(2), minus f(2) multiplied by g'(2), all divided by g(2) squared.
Step-by-step explanation:
To find the value of (fg)'(2), we need to use the product rule of differentiation. The product rule states that the derivative of the product of two functions is equal to the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function.
Using this rule, we can find that:
(fg)'(2) = f'(2) * g(2) + f(2) * g'(2)
Substituting the given values, we get:
(fg)'(2) = 10 * (-1) + 5 * 6
(fg)'(2) = -10 + 30
(fg)'(2) = 20
Therefore, (fg)'(2) = 20
To find the value of (f/g)'(2), we need to use the quotient rule of differentiation. The quotient rule states that the derivative of the quotient of two functions is equal to the derivative of the first function multiplied by the second function, minus the first function multiplied by the derivative of the second function, all divided by the square of the second function.
Using this rule, we can find that:
(f/g)'(2) = (f'(2) * g(2) - f(2) * g'(2)) / (g(2))^2
Substituting the given values, we get:
(f/g)'(2) = (10 * (-1) - 5 * 6) / (-1)^2
(f/g)'(2) = (-10 - 30) / 1
(f/g)'(2) = -40 / 1
(f/g)'(2) = -40
Therefore, (f/g)'(2) = -40