Natalie G. answered • 02/17/21

Math, Writing and Test Prep Tutor for All Ages, 11+ Years Experience

First off, I'm not sure why f(x) is stated in the problem? In any case, start with the product rule for derivatives. Let's designate x^{2 }+ 12 as d(x) to prevent confusion. Here's the product rule:

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

Take the derivative of d(x) using the power rule. We'll leave g'(x) and g(x) for now as unknowns.

h'(x)= (2x)g(x) + (x^{2 }+ 12)g'(x)

You're solving for h'(1).

h'(1)= (2)(1)(g(1)) + (1^{2}+12)g'(1)

Plug in what you were given in the original problem and simplify.

h'(1)= 2(-2) + (13)(3) = -4 + 39 = 35

h'(1) = 35