Let's analyze the image step by step. The table provides values for $f(x)$, $g(x)$, and their derivatives $\frac{d f}{d x}$ and $\frac{d g}{d x}$ at specific values of $x$. You are tasked with finding the derivative of $h(x)$ at certain points based on given functions of $f(x)$ and $g(x)$.

The provided formulas for $h(x)$ are:

1. $h(x) = -9f(x) + 9g(x)$, and you need to find $h'(2)$.
2. $h(x) = 5f(x) - 2g(x)$, and you need to find $h'(4)$.
3. $h(x) = 7f(x) - 2g(x)$, and you need to find $h'(1)$.

Approach:

The derivative of $h(x)$ can be computed using the following rule: $h'(x) = -9f'(x) + 9g'(x)$ for the first case, $h'(x) = 5f'(x) - 2g'(x)$ for the second case, $h'(x) = 7f'(x) - 2g'(x)$ for the third case.

We will then look at the values of $f'(x)$ and $g'(x)$ from the table at the respective points of $x = 2$, $x = 4$, and $x = 1$.

Solving each one:

1. For $h'(2)$: $h'(2) = -9f'(2) + 9g'(2)$ From the table, $f'(2) = -4$ and $g'(2) = -12$. $h'(2) = -9(-4) + 9(-12) = 36 - 108 = -72$

2. For $h'(4)$: $h'(4) = 5f'(4) - 2g'(4)$ From the table, $f'(4) = 4$ and $g'(4) = 0$. $h'(4) = 5(4) - 2(0) = 20$

3. For $h'(1)$: $h'(1) = 7f'(1) - 2g'(1)$ From the table, $f'(1) = 8$ and $g'(1) = -27$. $h'(1) = 7(8) - 2(-27) = 56 + 54 = 110$

Final Answers:

1. $h'(2) = -72$
2. $h'(4) = 20$
3. $h'(1) = 110$

Would you like further details or clarification on any of the steps?

Related Questions:

1. What is the chain rule for differentiation, and how is it applied here?
2. How would $h(x)$ change if it was a product of $f(x)$ and $g(x)$?
3. Can you compute the second derivative $h''(x)$ for any of these cases?
4. How would you approach this if the functions were not differentiable at certain points?
5. How do you interpret the derivative geometrically?

Tip:

When applying differentiation rules to composite functions, always verify whether you need to apply additional rules like the product or chain rule!