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, $\sqrt{x} = x^{1/2}$.