https://file.aiquickdraw.com/mathbot/file/1726029895129wfi3hdsx.jpg

Let's start with problem 32 and 36 from the image.

### Problem 32:  
\[ f(x) = -x^2 + 3x + 2 \]

We are tasked with finding the derivative \( f'(x) \), and then calculating \( f'(1) \), \( f'(2) \), and \( f'(3) \).

#### Step 1: Find the derivative
To differentiate \( f(x) = -x^2 + 3x + 2 \), we apply the power rule for each term:
- The derivative of \( -x^2 \) is \( -2x \)
- The derivative of \( 3x \) is \( 3 \)
- The derivative of the constant \( 2 \) is \( 0 \)

Thus, the derivative is:
\[ f'(x) = -2x + 3 \]

#### Step 2: Calculate \( f'(1) \), \( f'(2) \), and \( f'(3) \)
1. \( f'(1) = -2(1) + 3 = -2 + 3 = 1 \)
2. \( f'(2) = -2(2) + 3 = -4 + 3 = -1 \)
3. \( f'(3) = -2(3) + 3 = -6 + 3 = -3 \)

### Problem 36:
\[ f(x) = \frac{6}{x} - 2 \]

#### Step 1: Find the derivative
To differentiate \( f(x) = \frac{6}{x} - 2 \), we rewrite the term \( \frac{6}{x} \) as \( 6x^{-1} \):
- The derivative of \( 6x^{-1} \) using the power rule is \( -6x^{-2} \), which simplifies to \( -\frac{6}{x^2} \)
- The derivative of \( -2 \) is \( 0 \)

Thus, the derivative is:
\[ f'(x) = -\frac{6}{x^2} \]

#### Step 2: Calculate \( f'(1) \), \( f'(2) \), and \( f'(3) \)
1. \( f'(1) = -\frac{6}{1^2} = -6 \)
2. \( f'(2) = -\frac{6}{2^2} = -\frac{6}{4} = -\frac{3}{2} \)
3. \( f'(3) = -\frac{6}{3^2} = -\frac{6}{9} = -\frac{2}{3} \)

### Summary of Results:
- **Problem 32:**
  - \( f'(x) = -2x + 3 \)
  - \( f'(1) = 1 \)
  - \( f'(2) = -1 \)
  - \( f'(3) = -3 \)

- **Problem 36:**
  - \( f'(x) = -\frac{6}{x^2} \)
  - \( f'(1) = -6 \)
  - \( f'(2) = -\frac{3}{2} \)
  - \( f'(3) = -\frac{2}{3} \)

Would you like more details on any of these steps? 

Here are 5 related questions you could explore:
1. How do you apply the power rule for derivatives in general?
2. What is the quotient rule for derivatives, and when should it be used?
3. How can we differentiate functions with more complex exponents?
4. What is the significance of the derivative at specific points like \( f'(1), f'(2), f'(3) \)?
5. How can we visualize the behavior of \( f(x) \) by examining its derivative?

**Tip**: The power rule is the simplest method for differentiating polynomial terms, but for more complex expressions involving division, the quotient rule or chain rule might be necessary.

Learn how to differentiate the polynomial function f(x) = -x^2 + 3x + 2 and the rational function f(x) = 6/x - 2. This solution covers the application of the power rule and the differentiation of rational functions. It also calculates f'(1), f'(2), and f'(3) for both functions. Suitable for high school calculus students.

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