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We are tasked with finding \( G'(3) \) where the function \( G(x) \) is defined as:

\[
G(x) = 2g(x) - h(x) + 8
\]

To differentiate this function, apply the derivative:

\[
G'(x) = 2g'(x) - h'(x)
\]

Next, we use the table provided for \( x = 3 \):
- \( g'(3) = -15 \)
- \( h'(3) = 6 \)

Now substitute these values into the derivative expression for \( G'(x) \):

\[
G'(3) = 2(-15) - 6 = -30 - 6 = -36
\]

Thus, \( G'(3) = -36 \).

Would you like further clarification on this? Here are some related questions you might explore:

1. What is the physical interpretation of a derivative in this context?
2. How would \( G(x) \) behave for values of \( x \neq 3 \)?
3. How does adding a constant (like \( +8 \)) to a function affect its derivative?
4. What are the roles of \( g(x) \) and \( h(x) \) in defining the behavior of \( G(x) \)?
5. How would the second derivative \( G''(x) \) look in this case?

**Tip:** Remember, constants like \( +8 \) vanish when differentiating, as the derivative of a constant is zero.

The following table lists the value of functions g and h, and of their derivatives g' and h', for x = 3. Let function G be defined as G(x) = 2g(x) - h(x) + 8. What is G'(3)?

Learn how to differentiate the function G(x) = 2g(x) - h(x) + 8 using values from a given table. This problem explores the sum rule and constant rule of differentiation, and calculates G'(3). Suitable for high school students in Grades 11-12.

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