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) $-\frac{36}{25}$

              c) 4

Step-by-step 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:

$(\frac{f}{g})'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}$

$(\frac{f}{g})' (5)=\frac{f'(5)g(5)-f(5)g'(5)}{[g(5)]^2}$

$(\frac{f}{g})'(5)=\frac{6(-5)-3(2)}{(-5)^{2}}$

$(\frac{f}{g})'(5)=\frac{-36}{25}$

c) $(\frac{g}{f})'(5)=\frac{g'(5)f(5)-g(5)f'(5)}{[f(5)]^{2}}$

$(\frac{g}{f})'(5)=\frac{2(3)-(-5)(6)}{3^{2}}$

$(\frac{g}{f})'(5)=\frac{36}{9}$

$(\frac{g}{f})'(5)=4$