Suppose that f(5) = 1, f '(5) = 8, g(5) = −4, and g'(5) = 2. Find the following values.(a)    (fg)'(5) (b)    (f/g)'(5) (c)    (g/f)'(5)

(a) (fg)'(5)

This is the derivative of the product of two functions, which can be found using the product rule: (fg)' = f'g + fg'.

Substituting the given values, we get:

(fg)'(5) = f'(5)g(5) + f(5)g'(5) = 8*(-4) + 1*2 = -32 + 2 = -30

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

This is the derivative of the quotient of two functions, which can be found using the quotient rule: (f/g)' = (gf' - fg') / g^2.

Substituting the given values, we get:

(f/g)'(5) = (g(5)f'(5) - f(5)g'(5)) / (g(5))^2 = (-48 - 12) / (-4)^2 = -34 / 16 = -17/8

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

This is also the derivative of the quotient of two functions, which can be found using the quotient rule: (g/f)' = (fg' - gf') / f^2.

Substituting the given values, we get:

(g/f)'(5) = (f(5)g'(5) - g(5)f'(5)) / (f(5))^2 = (12 - -48) / 1^2 = 34.