We are asked to find $h'(2)$ for various functions $h(x)$, where the derivatives of the functions $f$ and $g$ are given as:

$f'(2) = -4, \quad g'(2) = -3$

The derivative of each function $h(x)$ will be calculated using the rules of differentiation, particularly the constant multiple rule and the sum rule for derivatives.

1. $h(x) = 8f(x)$

By the constant multiple rule, we have: $h'(x) = 8f'(x)$ Evaluating at $x = 2$: $h'(2) = 8f'(2) = 8 \times (-4) = -32$

2. $h(x) = -9g(x)$

Using the constant multiple rule again: $h'(x) = -9g'(x)$ Evaluating at $x = 2$: $h'(2) = -9g'(2) = -9 \times (-3) = 27$

3. $h(x) = 2f(x) + 5g(x)$

By the sum and constant multiple rules: $h'(x) = 2f'(x) + 5g'(x)$ Evaluating at $x = 2$: $h'(2) = 2f'(2) + 5g'(2) = 2 \times (-4) + 5 \times (-3) = -8 - 15 = -23$

4. $h(x) = 6g(x) - 9f(x)$

By the sum and constant multiple rules: $h'(x) = 6g'(x) - 9f'(x)$ Evaluating at $x = 2$: $h'(2) = 6g'(2) - 9f'(2) = 6 \times (-3) - 9 \times (-4) = -18 + 36 = 18$

5. $h(x) = 4f(x) + 3g(x) + 12$

By the sum and constant multiple rules: $h'(x) = 4f'(x) + 3g'(x) + 0$ (The constant 12 disappears after differentiation.) Evaluating at $x = 2$: $h'(2) = 4f'(2) + 3g'(2) = 4 \times (-4) + 3 \times (-3) = -16 - 9 = -25$

6. $h(x) = -6g(x) - 13f(x) + 8x$

By the sum and constant multiple rules: $h'(x) = -6g'(x) - 13f'(x) + 8$ Evaluating at $x = 2$: $h'(2) = -6g'(2) - 13f'(2) + 8 = -6 \times (-3) - 13 \times (-4) + 8 = 18 + 52 + 8 = 78$

Summary of Results:

1. $h'(2) = -32$
2. $h'(2) = 27$
3. $h'(2) = -23$
4. $h'(2) = 18$
5. $h'(2) = -25$
6. $h'(2) = 78$

Would you like more details or have any questions on the steps?

Here are some additional questions related to this topic:

1. How do the product and quotient rules for derivatives apply in more complex functions of $f(x)$ and $g(x)$?
2. Can you explain how chain rule would be used if the functions were compositions instead of linear combinations?
3. How would implicit differentiation affect the calculation of $h'(x)$ in these examples?
4. How do you handle differentiation when functions include higher powers of $f(x)$ and $g(x)$?
5. What happens to the derivative if instead of constants, there were functions multiplied with $f(x)$ and $g(x)$?

Tip: Always check if the function includes constants, as they disappear upon differentiation, simplifying your work.