Suppose that f(2) = −4, g(2) = 2, f '(2) = −5, and g'(2) = 1. Find h'(2).

Question

 h(x) = 5f(x) − 4g(x)

h'(2) =  

(b)    h(x) = f(x)g(x)

h'(2) =  

(c)    h(x) = f(x)

g(x)

h'(2) =  

(d)    h(x) = g(x)

1 + f(x)

h'(2) =  

Answer 1

When evaluating h'(2) it's important to make sure you are taking the derivative before evaluating.

a) h(x) = 5f(x) - 4g(x)

h'(x) = 5f'(x) - 4g'(x)

therefore

h'(2) = 5f'(2) - 4g'(2)

h'(2) = 5(-5) - 4(1) = -25-4 = -29

so

when h(x) = 5f(x) - 4g(x), h'(2) = -29

b) h(x) = f(x)g(x)

This is a product rule, so

h'(x) = f'(x)g(x) + g'(x)f(x)

and

h'(2) = f'(2)g(2) = g'(2)f(2)

h'(2) = (-5)(2) + (1)(-4) = -10 + -4 = -10 - 4 = -14

so

h'(2) = -14

c) & d) look like quotient rules, but it's hard to tell with the question. If it is a quotient rule, the derivative of c):

h(x) = f(x)/g(x)

h'(x) = (g(x)f'(x) - f(x)g'(x))/g(x)²

^thus

h'(2) = (g(2)f'(2) - f(2)g'(2))/(g(2)²

and

h'(2) = ((2)(-5) - (-4)(1))/(2²) = ( -10 - -4)/4 = (-10 + 4)/4 = -6/4 = -3/2

I'll leave the second quotient rule to be performed using the above formula.