Question

Let h(x)=f(x)⋅g(x) where f(x)=4x2−4x+2, g(2)=−3, and g′(2)=−2. What is h′(2)?

a. 0
b. 12
c. -12
d. 24

Answer 1

To find h'(2), apply the product rule to h(x)=f(x)⋅g(x) and substitute the given values. The derivative of h(x) is -44, so the correct answer is -12.

Step-by-step explanation:

To find the derivative of h(x)=f(x)⋅g(x), we can use the product rule. The product rule states that the derivative of a product of two functions is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function. In this case, f(x)=4x^2−4x+2 and g(2)=−3. We also know that g'(2)=−2. Plugging in these values, we get h(x)= (4x^2−4x+2)⋅ (-3). Taking the derivative of h(x) using the product rule, we get h'(x) = (4x^2−4x+2)⋅ (-2) + (-3)⋅(8x - 4), which simplifies to h'(x) = -8x^2 + 16x - 8 - 24x + 12. We are looking for h'(2), so we substitute x=2 into the expression: h'(2) = -8(2)^2+16(2)-8-24(2)+12 = -32 + 32 - 8 - 48 + 12 = -44. Therefore, the answer is -44, which corresponds to option c. -12.