https://file.aiquickdraw.com/mathbot/file/1724138086680d17qnqa9.jpg

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?

---

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.

Explore how to solve calculus problems involving derivatives and integrals step-by-step. Learn differentiation using the chain rule and product rule, and integration of polynomial functions. Suitable for advanced high school students and college-level calculus enthusiasts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation