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Let's go through the derivatives for each function step by step. The general rule I'll use for finding the derivative of a polynomial term \( ax^n \) is:

\[
\frac{d}{dx}(ax^n) = a \cdot n \cdot x^{n-1}
\]

### 1. \( y = 12x^3 - 8x^2 + 7x + 5 \)
- The derivative of \( 12x^3 \) is \( 36x^2 \).
- The derivative of \( -8x^2 \) is \( -16x \).
- The derivative of \( 7x \) is \( 7 \).
- The derivative of the constant \( 5 \) is \( 0 \).

So, \( y' = 36x^2 - 16x + 7 \).

---

### 2. \( y = 8x^3 - 5x^2 - \frac{x}{12} \)
- The derivative of \( 8x^3 \) is \( 24x^2 \).
- The derivative of \( -5x^2 \) is \( -10x \).
- The derivative of \( -\frac{x}{12} \) is \( -\frac{1}{12} \).

So, \( y' = 24x^2 - 10x - \frac{1}{12} \).

---

### 3. \( y = 3x^4 - 6x^3 + \frac{x^2}{8} + 5 \)
- The derivative of \( 3x^4 \) is \( 12x^3 \).
- The derivative of \( -6x^3 \) is \( -18x^2 \).
- The derivative of \( \frac{x^2}{8} \) is \( \frac{x}{4} \).
- The derivative of the constant \( 5 \) is \( 0 \).

So, \( y' = 12x^3 - 18x^2 + \frac{x}{4} \).

---

### 4. \( y = 5x^4 + 9x^3 + 12x^2 - 7x \)
- The derivative of \( 5x^4 \) is \( 20x^3 \).
- The derivative of \( 9x^3 \) is \( 27x^2 \).
- The derivative of \( 12x^2 \) is \( 24x \).
- The derivative of \( -7x \) is \( -7 \).

So, \( y' = 20x^3 + 27x^2 + 24x - 7 \).

---

### 5. \( f(x) = 6x^{3.5} + 10x^{0.5} \)
- The derivative of \( 6x^{3.5} \) is \( 21x^{2.5} \).
- The derivative of \( 10x^{0.5} \) is \( 5x^{-0.5} \).

So, \( f'(x) = 21x^{2.5} + 5x^{-0.5} \).

---

### 6. \( f(x) = -2x^{1.5} + 12x^{0.5} \)
- The derivative of \( -2x^{1.5} \) is \( -3x^{0.5} \).
- The derivative of \( 12x^{0.5} \) is \( 6x^{-0.5} \).

So, \( f'(x) = -3x^{0.5} + 6x^{-0.5} \).

---

### 7. \( y = 8\sqrt{x} + 6x^{3/4} \)
- \( \sqrt{x} = x^{1/2} \), so the derivative of \( 8\sqrt{x} \) is \( 4x^{-0.5} \).
- The derivative of \( 6x^{3/4} \) is \( \frac{9}{2}x^{-1/4} \).

So, \( y' = 4x^{-0.5} + \frac{9}{2}x^{-0.25} \).

---

### 8. \( y = -100\sqrt{x} - 11x^{2/3} \)
- The derivative of \( -100\sqrt{x} = -100x^{1/2} \) is \( -50x^{-0.5} \).
- The derivative of \( -11x^{2/3} \) is \( -\frac{22}{3}x^{-1/3} \).

So, \( y' = -50x^{-0.5} - \frac{22}{3}x^{-1/3} \).

---

### 9. \( y = 10x^{-3} + 5x^{-4} - 8x \)
- The derivative of \( 10x^{-3} \) is \( -30x^{-4} \).
- The derivative of \( 5x^{-4} \) is \( -20x^{-5} \).
- The derivative of \( -8x \) is \( -8 \).

So, \( y' = -30x^{-4} - 20x^{-5} - 8 \).

---

### 10. \( y = 5x^{-5} - 6x^{-2} + 13x^{-1} \)
- The derivative of \( 5x^{-5} \) is \( -25x^{-6} \).
- The derivative of \( -6x^{-2} \) is \( 12x^{-3} \).
- The derivative of \( 13x^{-1} \) is \( -13x^{-2} \).

So, \( y' = -25x^{-6} + 12x^{-3} - 13x^{-2} \).

---

Let me know if you'd like further details or clarification on any of the steps!

Here are 5 related questions you might find helpful:
1. What is the derivative of \( y = x^5 + 3x^4 - x^3 + 7x - 2 \)?
2. How do you find the second derivative of a function?
3. What is the chain rule for derivatives, and how does it work?
4. How would you find the derivative of a trigonometric function like \( \sin(x) \) or \( \cos(x) \)?
5. What is the derivative of a natural logarithm function \( \ln(x) \)?

**Tip:** When dealing with fractional exponents, it's helpful to rewrite roots as powers of \( x \). For example, \( \sqrt{x} = x^{1/2} \).

Find the derivative of each function defined as follows.

Explore detailed step-by-step solutions for finding derivatives of various polynomial and radical functions including power, product, and chain rules. Suitable for college or university level students learning calculus.

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