If f(1) = 3, g(1) = −4, f '(1) = 8, and g'(1) = −7, find each of the following.

Question

it's pretty straightforward, but for some reason my answers are wrong.

a) (f*g)''(1)

b) (f/g)'(1)

c) ((f/f-g)'(1)

These are my answers just in case I'm completely wrong.

a) 8 * -7 = -56

b) = -(8/7)

c) 8/(8-(-7) = 8/15

for c also tried 8/(8-7) =8/1=8

Answer 1

Question a) Since you are taking the derivative of two multiplied functions you must use the product rule which says (f•g)' = f•g' + f '•g and since you are told the value of each of these at x = 1, (f•g)'(1) = f(1)•g'(1) + f '(1)•g(1) = 3(-7) + 8(-4) = -21 - 32 = -53

Question b) Since you are taking the derivative of two divided functions you must use the quotient rule which says (f/g)' = (f '•g - fg')/g² so (f/g)'(1) = [(8)(-4) - (3)(-7)]/(-4)² = (-32 - -21)/16 = -11/16

Question c) You need to use both the quotient rule and the product rule. To make it less confusing, let me re-write them using "u" and "v" since there are already "f's" and "g's":

Product rule: (u•v)' = u'v + uv'

Quotient rule: (u/v)' = (u'v - v'u)/v²

In this case u = f and v = f•g

So lets do the denominator first.

v(1) = f(1)•g(1) = (3)(-4) = -12 and therefore v² = (-12)² = 144

v'(1) = f '(1)•g(1) + f(1)•g'(1) = (8)(-4) + (3)(-7) = -53

u(1) = f(1) = 3

u'(1) = f'(1) = 8

Using the quotient rule (u/v)' = (u'v - v'u)/v² = [(8)(-12) - (-53)(3)]/144 = (-96 - -159)/144 = 63/144 = 7/16