Let's go through each problem one by one to find the derivatives.

Problem 1:

$y = \frac{x^2 + 4}{x^3 - 1}$ This is a quotient, so we will use the quotient rule for derivatives. The quotient rule states: $\frac{d}{dx} \left(\frac{u(x)}{v(x)}\right) = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2}$ Here:

• $u(x) = x^2 + 4$
• $v(x) = x^3 - 1$

We find $u'(x)$ and $v'(x)$, then apply the quotient rule.

Problem 2:

$g(x) = (5x^2 - 2)(x^3 + 3x)$ This is a product, so we will use the product rule for derivatives: $\frac{d}{dx} \left(u(x)v(x)\right) = u'(x)v(x) + u(x)v'(x)$ Here:

• $u(x) = 5x^2 - 2$
• $v(x) = x^3 + 3x$

Problem 3:

$h(x) = 3 \cot(x) - 2 \cos(x)$ For this, we will use standard derivative formulas for trigonometric functions:

• Derivative of $\cot(x)$ is $-\csc^2(x)$
• Derivative of $\cos(x)$ is $-\sin(x)$

Problem 4:

$y = \frac{5 - \sqrt{5}}{5^2}$ This is a constant, so its derivative is $0$.

Problem 5:

$f(x) = (x^3 - 2x)(x^{-4} + x^{-2})$ We will use the product rule again for this problem.

Let me know if you'd like me to go into further detail for each step, or if you have any questions about these derivatives! Would you like to explore this further?

Here are 5 related questions you might find interesting:

1. How do you apply the product and quotient rules to functions with higher powers?
2. What is the chain rule and how does it simplify differentiation?
3. What are the derivative formulas for trigonometric functions?
4. How does differentiation of constants work?
5. How do you handle functions with negative exponents during differentiation?

Tip: When differentiating fractions or products, always check if you can simplify the function first to make the derivative process easier.