https://file.aiquickdraw.com/mathbot/file/1728053758707uv8ua10t.jpg

Let's break down the problems in the image and provide solutions:

1. **Problem 1: \( f(x) = 2x - 6x + 1 \), with \( c = 0 \)**  
   This function is a linear combination of terms involving \(x\). The problem asks to simplify and solve when \( c = 0 \).
   \[
   f(x) = 2x - 6 + 1 = 2x - 5
   \]
   Solving for \(x\) when \(f(x) = 0\):
   \[
   2x - 5 = 0 \implies 2x = 5 \implies x = \frac{5}{2}
   \]

2. **Problem 2: The derivative of \( f(x) = \frac{1}{x - 2} \)**  
   To differentiate this, we apply the quotient rule or rewrite the expression as \( f(x) = (x - 2)^{-1} \).
   \[
   f'(x) = - (x - 2)^{-2} \cdot 1 = -\frac{1}{(x - 2)^2}
   \]

3. **Problem 4: Find the derivative of \( y = \pi^5 \)**  
   Since \( y = \pi^5 \) is a constant (since \(\pi^5\) is just a number), its derivative is:
   \[
   y' = 0
   \]

4. **Problem 7: Find the derivative of \( y = \frac{8}{3x^2} \)**  
   Rewrite \( y \) as \( y = \frac{8}{3}x^{-2} \). Now, apply the power rule for differentiation:
   \[
   y' = \frac{8}{3} \cdot (-2) x^{-3} = -\frac{16}{3x^3}
   \]

5. **Problem 10: Find the derivative of \( y = \frac{4x^2 - 5x + 6}{3} \)**  
   Since the entire expression is divided by 3, we can factor out the constant and differentiate:
   \[
   y' = \frac{1}{3} \cdot \left( 8x - 5 \right) = \frac{8x - 5}{3}
   \]

Let me know if you'd like further details or if any step needs more clarification. Here are some related questions you can explore:

1. How do you apply the chain rule in more complex derivatives?
2. How would you differentiate implicit functions?
3. Can you explain how the quotient rule is applied to rational functions?
4. How do higher-order derivatives work, and what do they represent?
5. How do you find critical points using the first derivative?

**Tip:** Always simplify functions before differentiating when possible to avoid unnecessary complexity in calculations.

Find the derivative of various functions as shown in the image.

Explore how to differentiate functions using the Power and Quotient Rules. This calculus problem set involves finding derivatives for constants, polynomial expressions, and rational functions. Ideal for students at the high school or college level.

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