Let's go through the questions one by one.

Question Four: Find the derivatives of the following

1. $y = \sin^2(4x + \frac{\pi}{3})$
   To find the derivative $\frac{dy}{dx}$, we can use the chain rule.

   $y = \sin^2(u) \quad \text{where} \quad u = 4x + \frac{\pi}{3}$ First, find the derivative of $\sin^2(u)$ with respect to $u$: $\frac{dy}{du} = 2\sin(u)\cos(u) = \sin(2u)$ Next, differentiate $u = 4x + \frac{\pi}{3}$ with respect to $x$: $\frac{du}{dx} = 4$ Combine these using the chain rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \sin(2u) \cdot 4 = 4\sin(2(4x + \frac{\pi}{3}))$ Simplifying further: $\frac{dy}{dx} = 4\sin(8x + \frac{2\pi}{3})$

2. $y = (2x^3 + 4x^2 + 5)^{10}$
   Again, using the chain rule: $y = (v)^{10} \quad \text{where} \quad v = 2x^3 + 4x^2 + 5$ Differentiate: $\frac{dy}{dv} = 10v^9$ Then: $\frac{dv}{dx} = 6x^2 + 8x$ Combine these: $\frac{dy}{dx} = 10(2x^3 + 4x^2 + 5)^9 \cdot (6x^2 + 8x)$

3. $y = x^2 e^{3x} + 5$
   Use the product rule for $x^2 e^{3x}$ and differentiate 5 (which is constant): $\frac{dy}{dx} = 2x e^{3x} + x^2 \cdot 3e^{3x} = 2xe^{3x} + 3x^2 e^{3x}$ The derivative is: $\frac{dy}{dx} = e^{3x}(2x + 3x^2)$

Question Five

1. Calculate $\frac{dy}{dx}$ for the following functions:

   a) $y = 5x^2 - \frac{2x^3}{5} + 3$
   Differentiate each term: $\frac{dy}{dx} = 10x - \frac{6x^2}{5}$

   b) $y = 12x^3 - 4x^2 + 10$
   Differentiate each term: $\frac{dy}{dx} = 36x^2 - 8x$

2. Integrate the following functions with respect to $x$:

   a) $y = 3x^2 - 3x^4 + 8$
   Integrate term by term: $\int y \, dx = x^3 - \frac{3x^5}{5} + 8x + C$

   b) $y = 2x^3 - x^2 + 4x + 1$
   Integrate term by term: $\int y \, dx = \frac{2x^4}{4} - \frac{x^3}{3} + 2x^2 + x + C = \frac{x^4}{2} - \frac{x^3}{3} + 2x^2 + x + C$

Would you like further details on any specific part or have any questions?

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Here are 5 related questions that could expand your understanding:

1. How do you apply the product rule when differentiating a product of three functions?
2. What is the significance of the constant $C$ in indefinite integrals?
3. How can we use integration to find the area under a curve?
4. What are some real-world applications of derivatives?
5. How do higher-order derivatives relate to the original function?

Tip: When solving complex derivatives or integrals, breaking down the problem into smaller parts can make the process easier to manage.