Suppose that

f(5) = 3, f '(5) = 4, g(5) = −6, and g'(5) = 1.
Find the following values.
(a)
(fg)'(5)

(b)
(f/g)'(5)

(c)
(g/f)'(5)

To find the values (a), (b), and (c), we need to use the product rule and the quotient rule of differentiation.

(a) To find (fg)'(5), we can use the product rule, which states that the derivative of the product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function.

So, we have (fg)'(5) = f'(5) * g(5) + f(5) * g'(5)

Substituting the given values, we get (fg)'(5) = 4 * (-6) + 3 * 1
= -24 + 3
= -21

Therefore, (fg)'(5) = -21.

(b) To find (f/g)'(5), we can use the quotient rule, 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, we have (f/g)'(5) = (f'(5) * g(5) - f(5) * g'(5)) / (g(5))^2

Substituting the given values, we get (f/g)'(5) = (4 * (-6) - 3 * 1) / (-6)^2
= (-24 - 3) / 36
= -27 / 36
= -3/4

Therefore, (f/g)'(5) = -3/4.

(c) To find (g/f)'(5), we can again use the quotient rule.

So, we have (g/f)'(5) = (g'(5) * f(5) - g(5) * f'(5)) / (f(5))^2

Substituting the given values, we get (g/f)'(5) = (1 * 3 - (-6) * 4) / (3)^2
= (3 + 24) / 9
= 27 / 9
= 3

Therefore, (g/f)'(5) = 3.