Suppose that f(5) = 1, f '(5) = 5, g(5) = −4, and g'(5) = 7. Find the following values.

Question

Suppose that f(5) = 1, f '(5) = 5, g(5) = −4, and g'(5) = 7.

Find the following values.

(a)    (fg)'(5)

 

(b)    (f/g)'(5)

 

(c)    (g/f)'(5)

Answer 1

a) By the Product Rule, (fg)' = f'g + g'f. Therefore, (fg)'(5) = f'(5)g(5) + g'(5)f(5) = 5(-4) + 7(1) = -20 + 7 = -13.

b) By the Quotient Rule, (f/g)' = (f'g - g'f)/g². Therefore, (f/g)'(5) = (f'(5)g(5) - g'(5)f(5))/(g(5))² = (5(-4) - 7(1))/((-4)^2) = (-20 - 7)/16 = -27/16.

c) By the Quotient Rule, (g/f)' = (g'f - f'g)/f². Therefore, (g/f)'(5) = (g'(5)f(5) - f'(5)g(5))/(f(5))^2 = (7(1) - 5(-4))/1² =(7 + 20)/1 = 27.