https://file.aiquickdraw.com/mathbot/file/1730636970579le37t3va.pdf

Let's go through each derivative problem in the file step-by-step:

### 1. \( f(x) = \frac{2x - 3}{3x + 4} \)
   - Find \( f'(x) \), \( f'(1) \), and \( f'\left(-\frac{4}{3}\right) \) if possible.

### 2. \( f(x) = x^{\frac{1}{3}} \)
   - Find \( f'(x) \), \( f'(1) \), and \( f'(0) \) if possible.

### 3. \( f(x) = \sqrt{x^{23}} \)
   - Simplify and find \( f'(x) \).

### 4. \( f(x) = (5x^2 - 5)(4x + 2) \)
   - Use the product rule to find \( f'(x) \).

### 5. \( f(x) = \frac{x^3 + 1}{x^2 + 2(x^2 - 2x + 1)} \)
   - Differentiate the function.

### 6. \( y = \left(\frac{3x^3 - 5x}{4x + 17}\right)^3 \)
   - Use the chain rule and quotient rule to find \( y' \).

### 7. \( y = \frac{\sqrt{x^2 + 1}}{1 + \sqrt{x^2 + 1}} \)
   - Differentiate using the quotient and chain rules.

### 8. \( y = x \sin x \)
   - Use the product rule to find \( y' \).

### 9. \( y = \frac{1 - \cos x}{1 + \cos x} \)
   - Differentiate using the quotient rule.

### 10. \( y = \frac{\sin x \sec x}{1 + \tan x} \)
   - Use trigonometric identities and the quotient rule to find \( y' \).

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Would you like a detailed solution for each question or any specific ones? Here are related questions to consider:

1. How to apply the quotient rule in more complex expressions?
2. What are some simplification techniques after differentiation?
3. How does the chain rule apply to functions with nested functions?
4. What are common mistakes when differentiating trigonometric functions?
5. How can you check the correctness of your derivative answer?

**Tip:** Always check if functions can be simplified before applying differentiation rules—it can make the process easier.

Solve each derivative problem as given:
1. \( f(x) = \frac{2x - 3}{3x + 4} \) - Find \( f'(x) \), \( f'(1) \), and \( f'\left(-\frac{4}{3}\right) \) if possible.
2. \( f(x) = x^{\frac{1}{3}} \) - Find \( f'(x) \), \( f'(1) \), and \( f'(0) \) if possible.
3. \( f(x) = \sqrt{x^{23}} \) - Simplify and find \( f'(x) \).
4. \( f(x) = (5x^2 - 5)(4x + 2) \) - Use the product rule to find \( f'(x) \).
5. \( f(x) = \frac{x^3 + 1}{x^2 + 2(x^2 - 2x + 1)} \) - Differentiate the function.
6. \( y = \left(\frac{3x^3 - 5x}{4x + 17}\right)^3 \) - Use the chain rule and quotient rule to find \( y' \).
7. \( y = \frac{\sqrt{x^2 + 1}}{1 + \sqrt{x^2 + 1}} \) - Differentiate using the quotient and chain rules.
8. \( y = x \sin x \) - Use the product rule to find \( y' \).
9. \( y = \frac{1 - \cos x}{1 + \cos x} \) - Differentiate using the quotient rule.
10. \( y = \frac{\sin x \sec x}{1 + \tan x} \) - Use trigonometric identities and the quotient rule to find \( y' \).

Explore detailed solutions to complex derivative problems involving quotient, product, and chain rules. Learn to differentiate functions like \( \frac{2x - 3}{3x + 4} \), \( x^{1/3} \), and trigonometric expressions. Suitable for advanced calculus students.

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