let f and g be functions that satisfy f'(2)=-4 and g'(2)=-3. find h'(2) for each function of h below. 1: h(x)=8f(x) 2:h(x)=-9g(x) 3: hx= 2f(x)+5g(x) 4:h(x)=6g(x)-9f(x) 5: h(x)=4f(x)+3g(x)+12 6: h(x)=-6g(x)-13f(x)+8x

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.

Let f and g be functions that satisfy f'(2) = -4 and g'(2) = -3. Find h'(2) for each function of h below: 1: h(x) = 8f(x), 2: h(x) = -9g(x), 3: h(x) = 2f(x) + 5g(x), 4: h(x) = 6g(x) - 9f(x), 5: h(x) = 4f(x) + 3g(x) + 12, 6: h(x) = -6g(x) - 13f(x) + 8x.

This page explains how to find h'(2) for various functions of f(x) and g(x) given f'(2) = -4 and g'(2) = -3. The solution uses differentiation rules like the constant multiple rule and sum rule, making it suitable for high school students in Grades 11-12.

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