Question

Suppose that f(5) = 3, f '(5) = 6, g(5) = −5, and g'(5) = 2. Find the following values. (a) (fg)'(5) (b) f g '(5) (c) g f '(5)

Answer 1

Answer: a) -24

b)
`-(36)/(25)`

c) 4

Explanation:

a) To determine the value of (fg)', use the product rule of derivative, i.e.:

(fg)'(x) = f'(x)g(x) + f(x)g'(x)

(fg)'(5) = f'(5)g(5) + f(5)g'(5)

(fg)'(5) = 6(-5) + 3(2)

(fg)'(5) = -24

b) The value is given by the use of the quotient rule of derivative:

`((f)/(g))'(x)=(f'(x)g(x)-f(x)g'(x))/([g(x)]^2)`

`((f)/(g))' (5)=(f'(5)g(5)-f(5)g'(5))/([g(5)]^2)`

`((f)/(g))'(5)=(6(-5)-3(2))/((-5)^(2))`

`((f)/(g))'(5)=(-36)/(25)`

c)
`((g)/(f))'(5)=(g'(5)f(5)-g(5)f'(5))/([f(5)]^(2))`

`((g)/(f))'(5)=(2(3)-(-5)(6))/(3^(2))`

`((g)/(f))'(5)=(36)/(9)`

`((g)/(f))'(5)=4`