Let's solve the differentiation problems given in the image.

Problem 11: $y = (1 + x^3)(3 - 2x)^{5/2}$

We will differentiate using the product rule and chain rule.

Let:

• $u = 1 + x^3$
• $v = (3 - 2x)^{5/2}$

Using the product rule: $\frac{dy}{dx} = u'v + uv'$

• $u' = \frac{d}{dx}(1 + x^3) = 3x^2$
• $v' = \frac{d}{dx} \left( (3 - 2x)^{5/2} \right)$
  • Apply the chain rule: $v' = \frac{5}{2}(3 - 2x)^{3/2} \cdot (-2) = -5(3 - 2x)^{3/2}$

Now, apply the product rule: $\frac{dy}{dx} = (3x^2)(3 - 2x)^{5/2} + (1 + x^3)(-5(3 - 2x)^{3/2})$

Problem 12: $y = \sqrt[3]{x^2 - 4x^{-3}}$

Rewrite this as: $y = (x^2 - 4x^{-3})^{1/3}$ Now, apply the chain rule.

Let:

• $u = x^2 - 4x^{-3}$
• $y = u^{1/3}$

Using the chain rule: $\frac{dy}{dx} = \frac{1}{3}u^{-2/3} \cdot u'$

• $u' = \frac{d}{dx} \left( x^2 - 4x^{-3} \right) = 2x + 12x^{-4}$

Now substitute into the chain rule: $\frac{dy}{dx} = \frac{1}{3}(x^2 - 4x^{-3})^{-2/3} \cdot (2x + 12x^{-4})$

Would you like further clarifications or detailed steps?

Here are some related questions you might explore:

1. How does the product rule work in differentiation?
2. How does the chain rule apply when differentiating functions with exponents?
3. What is the geometric interpretation of the derivative of these functions?
4. How would the process differ if higher derivatives were requested?
5. How do these derivative rules apply in solving optimization problems?

Tip: Always check if a function can be simplified before differentiating to make the process easier.