https://file.aiquickdraw.com/mathbot/file/1727415906434s0yvlr0y.jpg

Let's find the first and second derivatives for each function given in the image. I'll start with the first few and proceed step-by-step.

### 1. \( y = -x^2 + 3 \)
**First Derivative:**  
\[
y' = \frac{d}{dx}(-x^2 + 3) = -2x
\]

**Second Derivative:**  
\[
y'' = \frac{d}{dx}(-2x) = -2
\]

---

### 2. \( y = x^2 + x + 8 \)
**First Derivative:**  
\[
y' = \frac{d}{dx}(x^2 + x + 8) = 2x + 1
\]

**Second Derivative:**  
\[
y'' = \frac{d}{dx}(2x + 1) = 2
\]

---

### 3. \( s = 5t^3 - 3t^5 \)
**First Derivative:**  
\[
s' = \frac{d}{dt}(5t^3 - 3t^5) = 15t^2 - 15t^4
\]

**Second Derivative:**  
\[
s'' = \frac{d}{dt}(15t^2 - 15t^4) = 30t - 60t^3
\]

---

### 4. \( w = 3z^7 - 7z^3 + 21z^2 \)
**First Derivative:**  
\[
w' = \frac{d}{dz}(3z^7 - 7z^3 + 21z^2) = 21z^6 - 21z^2 + 42z
\]

**Second Derivative:**  
\[
w'' = \frac{d}{dz}(21z^6 - 21z^2 + 42z) = 126z^5 - 42z + 42
\]

---

Would you like me to continue with the remaining functions? If you have any specific function you'd like me to prioritize, please let me know.

### Additional Questions:
1. How would the derivatives change if we added an exponential term to each function?
2. What is the geometric interpretation of the first and second derivatives?
3. Can you identify points of inflection for these functions?
4. How do the first derivatives relate to the slopes of tangent lines?
5. What are some practical applications of the second derivative in physics?

### Tip:
For polynomial functions, the power rule is the most efficient way to find derivatives.

Find the first and second derivatives for the set of functions listed in the provided image.

Detailed step-by-step solutions for finding the first and second derivatives of various polynomial functions, including applications of the power rule and sum/difference rule. Explore how to differentiate complex expressions involving powers, constants, and sums. Suitable for advanced high school and college-level calculus.

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