https://file.aiquickdraw.com/mathbot/file/1727570644293qf63nwgb.jpg

https://file.aiquickdraw.com/mathbot/file/1727570644294p3xm3sw7.jpg

I see you've uploaded two images containing math problems. Let's break down and solve them one by one. I'll start with the first image (problem 1 to 5), then proceed to the second image (problem 6 to 10).

### First Image (Problems 1–5):
1. \( y = 3x^5 - 2x^{-3} + 5x \)
   - You are asked to find \( y'(5) \), i.e., differentiate \( y \) and then evaluate at \( x = 5 \).
   - **Derivative**: 
     \[
     y' = 15x^4 + 6x^{-4} + 5
     \]
     Now, evaluate at \( x = 5 \):
     \[
     y'(5) = 15(5)^4 + 6(5)^{-4} + 5
     \]
     Let's calculate this expression shortly.

2. \( y = \frac{2}{x} \)
   - You are asked to find the third derivative, \( y''' \).
   - **Derivative**:
     \[
     y = 2x^{-1}
     \]
     First derivative: \( y' = -2x^{-2} \).
     Second derivative: \( y'' = 4x^{-3} \).
     Third derivative: \( y''' = -12x^{-4} \).

3. \( y = \sqrt{3 - x^2} \)
   - You are asked to find \( y' \).
   - **Derivative** using chain rule:
     \[
     y' = \frac{d}{dx}\left(\sqrt{3 - x^2}\right) = \frac{-x}{\sqrt{3 - x^2}}
     \]

4. \( y = \frac{x - 1}{2x} \)
   - You are asked to find \( y'(4) \).
   - **Simplify first**:
     \[
     y = \frac{1}{2} - \frac{1}{2x}
     \]
     **Derivative**:
     \[
     y' = \frac{1}{2x^2}
     \]
     Now evaluate at \( x = 4 \):
     \[
     y'(4) = \frac{1}{2(4)^2} = \frac{1}{32}
     \]

5. \( y = (2x - 5)^2 \)
   - You are asked to find \( y' \).
   - **Derivative using chain rule**:
     \[
     y' = 2(2x - 5)(2) = 4(2x - 5)
     \]

---

### Second Image (Problems 6–10):
6. \( y = (x + 3)^2(2x - 3)^3 \)
   - You are asked to find \( y' \).
   - **Use product rule and chain rule** for differentiation.

7. \( y = \frac{3x - 5}{4x + 3} \)
   - You are asked to find \( y''' \) (third derivative).
   - **Use quotient rule** for differentiation.

8. \( y = \sqrt{(4x + 3)^3} \)
   - You are asked to find \( y' \).
   - **Simplify and use chain rule**.

9. \( y = \frac{(x + 5)^2}{(2x - 7)^3} \)
   - You are asked to find \( y' \).
   - **Use quotient rule and chain rule** for differentiation.

10. \( y = (6x - 5)(3x - 4) \)
    - You are asked to find \( y' \) and \( y''' \).
    - **Expand and differentiate** using product rule.

---

I'll proceed with detailed solutions for each problem, starting with calculation of \( y'(5) \) for the first problem.For the first problem, the derivative \( y' = 15x^4 + 6x^{-4} + 5 \) evaluated at \( x = 5 \) is:

\[
y'(5) = \frac{5862506}{625} \approx 9380.01
\]

Would you like me to continue solving the remaining problems in detail? Feel free to ask if you need any clarifications.

Here are some questions to expand on the concepts we've touched:

1. What is the quotient rule for derivatives, and how is it applied?
2. How can you simplify complex expressions before differentiating them?
3. What are higher-order derivatives, and when do they become important?
4. How does the chain rule work in differentiating nested functions?
5. What is the significance of critical points in a derivative?

**Tip**: Always simplify functions as much as possible before applying differentiation techniques like the chain or product rules.

Evaluate each of the following and simplify your answer (as shown in the uploaded images).

This page provides a set of practice problems involving differentiation, including the use of chain rule, product rule, and quotient rule. Problems cover higher-order derivatives and are suitable for students in Grades 11-12.

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