The image contains several calculus problems that ask for the derivatives of functions with respect to $x$. Let's go through each one and find the derivatives:

1. $y = 10x^3 + 8$

   • The derivative of $y$ with respect to $x$ is: $\frac{dy}{dx} = 30x^2$

2. $y = 10x^2 - 4x - 8$

   • The derivative of $y$ with respect to $x$ is: $\frac{dy}{dx} = 20x - 4$

3. $y = 10x^{0.5} + 4x^{-1} + 8$

   • The derivative of $y$ with respect to $x$ is: $\frac{dy}{dx} = 5x^{-0.5} - 4x^{-2}$

4. $y = 10x^2 + 4xz + 8$

   • Here, we treat $z$ as a constant, so the derivative of $y$ with respect to $x$ is: $\frac{dy}{dx} = 20x + 4z$

If you would like more details or explanations, feel free to ask! Would you like to explore further? Here are some questions to extend the discussion:

1. How do you find the derivative of a constant term?
2. What happens if $z$ is treated as a variable instead of a constant in the last equation?
3. Can you explain the power rule for differentiation?
4. What are the rules for differentiating terms like $x^{-1}$?
5. How would you approach higher-order derivatives (like second or third derivatives) for these functions?

Tip: Always remember that the derivative of a constant is zero!