Solve for h'(2)

Question

Suppose that f(2) = −4, g(2) = 3, f'(2) = −2, and g' (2) = 1.

 Find h''(2).

(a)  h(x) = 5 f(x) − 2g(x)

h''(2) = 

(b)  h(x) = f(x)g(x)

h''(2) = 

(c) h(x) = f(x) / 1+ f(x)

h' (2) =

(d) h(x) = g(x) / 1+ f(x)

Answer 1

Solve for h'(2)

Suppose that f(2) = −4, g(2) = 3, f'(2) = −2, and g' (2) = 1.

 Find h''(2).

Hello Aurora,

I am going to make two assumptions. First, that you are just wanting to see how to do part (c), since you are asking to solve for h'(2). Secondly, for part (c), that the fraction for h(x), in its original print, has 1 +f(x) as the entire denominator, in which case you want to type it as

(c) h(x) = f(x) / [1 + f(x)]

Making the second assumption, I am going to apply the Quotient Rule for Derivatives

→ h'(x) = {[1 + f(x)] * f '(x) - f(x) * f '(x)} / [1 + f(x)]²

→ h'(2) = {[1 + f(2)] * f '(2) - f(2) * f '(2)} / [1 + f(2)]²

→ h'(2) = {[1 + (-4)] * (-2) - (-4) * (-2)} / [1 + (-4)]²

→ h'(2) = {(-3) * (-2) - (8)} / (9)

→ h'(2) = -2 / 9

Were you looking for parts (a), (b), and (d) as well? If so, just let me know.

Thank you for posting the question.

Michael E.