https://file.aiquickdraw.com/mathbot/file/1726974713491ibyxzz4r.jpg

You uploaded an image with several calculus problems involving derivatives. Let's go over the solutions for each of them:

### 1. \( \frac{d}{dx} (x) \)
- The derivative of \( x \) is:
  \[
  \frac{d}{dx}(x) = 1
  \]

### 2. \( y = x^3 - x + 9 \)
- Differentiate each term:
  \[
  \frac{dy}{dx} = 3x^2 - 1
  \]

### 3. \( y = 4x^2 + 5x - \frac{1}{2} \)
- The derivative of a constant like \( -\frac{1}{2} \) is zero:
  \[
  \frac{dy}{dx} = 8x + 5
  \]

### 4. \( \frac{d}{dx} \left( (3x^5 - 2x^3 + 1)(x^4 - \frac{2}{3}x^3 + 5) \right) \)
- Use the product rule: \( \frac{d}{dx}(uv) = u'v + uv' \).
  - Let \( u = 3x^5 - 2x^3 + 1 \), so \( u' = 15x^4 - 6x^2 \)
  - Let \( v = x^4 - \frac{2}{3}x^3 + 5 \), so \( v' = 4x^3 - 2x^2 \)

The derivative will be:
  \[
  \frac{d}{dx} = (15x^4 - 6x^2)(x^4 - \frac{2}{3}x^3 + 5) + (3x^5 - 2x^3 + 1)(4x^3 - 2x^2)
  \]
  Simplify the expression after multiplying the terms.

### 5. \( y = x^9 - 5x^2 + 16 \)
- Differentiate term by term:
  \[
  \frac{dy}{dx} = 9x^8 - 10x
  \]

### 6. (This is the same as number 5, so the result will be the same.)

### 7. \( y = \frac{5}{x^9} \)
- Rewrite the function as \( y = 5x^{-9} \) and differentiate:
  \[
  \frac{dy}{dx} = -45x^{-10}
  \]

### 8. \( \frac{d}{dx} \left( \frac{2}{3}x^3 + \frac{1}{2}x^2 + 2x - \frac{5}{6} \right) \)
- Differentiate each term:
  \[
  \frac{d}{dx} = 2x^2 + x + 2
  \]

### 9. \( \frac{d}{dx} \left( \frac{5}{x^9} \right) \)
- Rewrite as \( 5x^{-9} \) and differentiate:
  \[
  \frac{d}{dx} = -45x^{-10}
  \]

### 10. \( y = \frac{8}{x^8} + \frac{x}{8} \)
- Rewrite the first term as \( 8x^{-8} \) and differentiate:
  \[
  \frac{dy}{dx} = -64x^{-9} + \frac{1}{8}
  \]

### 11. \( \frac{d}{dx} \left( \frac{5}{7}x^7 - \frac{5}{6}x^5 + 7x - 1 \right) \)
- Differentiate each term:
  \[
  \frac{d}{dx} = 5x^6 - \frac{25}{6}x^4 + 7
  \]

### 12. \( y = 2x^3 \left( 1 + \frac{1}{x^3} \right) \)
- Simplify the function first: 
  \[
  y = 2x^3 \cdot \left( \frac{x^3 + 1}{x^3} \right) = 2(x^3 + 1)
  \]
- Now, differentiate:
  \[
  \frac{dy}{dx} = 6x^2
  \]

### 13. \( y = \sqrt{x} \left( \sqrt[3]{x} + x \right) \)
- First simplify and apply the product rule:
  - Rewrite \( \sqrt{x} \) as \( x^{1/2} \) and \( \sqrt[3]{x} \) as \( x^{1/3} \)
  \[
  y = x^{1/2} \left( x^{1/3} + x \right)
  \]
  Differentiate using the product rule.

### 14. \( \frac{d}{dx} \left( \frac{1}{\sqrt[3]{x}} \right) \)
- Rewrite as \( x^{-1/3} \) and differentiate:
  \[
  \frac{d}{dx} = -\frac{1}{3}x^{-4/3}
  \]

### 15. \( y = (x^2 - 3x + 1)(x^4 - 5x^2 - x + 9) \)
- Use the product rule:
  - Let \( u = x^2 - 3x + 1 \) and \( v = x^4 - 5x^2 - x + 9 \)
  - Differentiate each and apply the product rule.

Would you like detailed steps for any of these problems, or explanations for any specific rules used?

### Further Questions:
1. Can you explain the product rule in detail with an example?
2. How do you handle the chain rule when dealing with square roots?
3. What is the best approach for differentiating polynomials with large exponents?
4. How can logarithmic differentiation simplify problems like question 12?
5. Can you clarify the difference between the quotient rule and rewriting as negative exponents?

### Tip:
When solving derivatives of products or quotients, it's helpful to first simplify the expressions as much as possible. This can often make the differentiation process much quicker and less error-prone.

Evaluate the derivative of the following functions.

This exercise involves solving derivatives of polynomial, rational, and composite functions using key calculus rules such as the power rule, product rule, and quotient rule. Step-by-step differentiation for functions such as x^3 - x + 9 and more complex products like (3x^5 - 2x^3 + 1)(x^4 - 2/3x^3 + 5).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation