Calc 1 problem

Question

Suppose that f(2) = 5, f '(2) = 5, g(2) = -6, and g'(2) = 4. Find the following values:  (fg)'(2);  (f/g)'(2); (g/f)'(2)

Please give reasons why or show work. Thanks!

Answer 1

Given f(2) = 5, g(2) = -6 , f'(2) = 5, g'(2) = 4

To find the required values we can use product and quotient rules in derivatives.

(fg)'(2) = d/dx(fg)(2)

d/dx(uv) = uv'+vu'

here u = f, v = g

d/dx(fg)(2) = f(2)g'(2)+g(2)f'(2) = 5*4+(-6)5 = 20-30 = -10

d/dx(u/v) = (vu'-uv')/v^2

Here u = f,v = g

d/dx(f/g)' = (gf'-fg')/g^2

= (-6*5-5*4)/(-6)^2

(f/g)'(2) = (-30-20)/36 = -50/36 = 25/18

(g/f)'(2) = d/dx(g/f)(2)

u = g, v = f

(g/f)'(2) = (fg'-gf')/f^2

= (5*4-(-6)*5)/5^2 = (20+30)/25 = 50/25 = 2

Comments

(f/g)'(2) = (-30-20)/36 = -50/36 = -25/18