Suppose that f(5) = 1, f '(5) = 4, g(5) = −2, and g'(5) = 7. Find the following values.

Question

(a)    (fg)'(5)

(b)    (f/g)'(5)

(c)    (g/f)'(5)

Answer 1

(a)

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

(fg)'(5) = f'(5)g(5) + f(5)g'(5) = 4*-2 + 7*1 = 1

(b)

(f/g)'(x) = ( f'(x)g(x) - f(x)g'(x) ) /g(x)^2 thus

(f/g)'(5) = ( f'(5)g(5) - f(5)g'(5) ) /g(5)^2 = (4*-2 - 7*1)/(-2)^2 = -15/4

(c)

(g/f)'(x) = ( g'(x)f(x) - g(x)f'(x) ) /f(x)^2 thus

(g/f)'(5) = ( g'(5)f(5) - g(5)f'(5) ) /f(5)^2 = (7*1 - 4*-2)/(1)^2 = 15