Final answer:
To find (f g)ʹ(6), use the product rule and substitute the given values to evaluate the derivative. To find (f/g)ʹ(6), use the quotient rule and substitute the given values to evaluate the derivative.
Step-by-step explanation:
(a) To find (f g)ʹ(6), we can use the product rule of differentiation, which states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. So, (f g)ʹ(6) = fʹ(6)g(6) + f(6)gʹ(6). Substituting the given values, we get (f g)ʹ(6) = 6*(-1) + 5*7 = -6 + 35 = 29.
(b) To find (f/g)ʹ(6), we can use the quotient rule of differentiation, which states that the derivative of the quotient of two functions is equal to the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. So, (f/g)ʹ(6) = (fʹ(6)g(6) - f(6)gʹ(6)) / (g(6))^2. Substituting the given values, we get (f/g)ʹ(6) = (6*(-1) - 5*7) / (-1)^2 = (-6 - 35) / 1 = -41.