https://file.aiquickdraw.com/mathbot/file/17267037895964f4xi4t1.jpg

To differentiate the given function \( y = (4x - 2)^2 (3 - x^5)^2 \), we can apply both the **product rule** and the **chain rule**.

### Step-by-Step Solution:

1. **Product Rule**: The function is a product of two terms, \( (4x - 2)^2 \) and \( (3 - x^5)^2 \). So, we apply the product rule:
   \[
   \frac{dy}{dx} = \frac{d}{dx} \left( (4x - 2)^2 \right) \cdot (3 - x^5)^2 + (4x - 2)^2 \cdot \frac{d}{dx} \left( (3 - x^5)^2 \right)
   \]

2. **Differentiate the first term** \( (4x - 2)^2 \) using the chain rule:
   - Let \( u = 4x - 2 \), then:
   \[
   \frac{d}{dx} \left( u^2 \right) = 2u \cdot \frac{du}{dx} = 2(4x - 2) \cdot 4 = 8(4x - 2)
   \]

3. **Differentiate the second term** \( (3 - x^5)^2 \) using the chain rule:
   - Let \( v = 3 - x^5 \), then:
   \[
   \frac{d}{dx} \left( v^2 \right) = 2v \cdot \frac{dv}{dx} = 2(3 - x^5) \cdot (-5x^4) = -10x^4(3 - x^5)
   \]

4. **Combine using the product rule**:
   - The derivative is:
   \[
   \frac{dy}{dx} = 8(4x - 2)(3 - x^5)^2 + (4x - 2)^2(-10x^4)(3 - x^5)
   \]

### Final Answer:
\[
\frac{dy}{dx} = 8(4x - 2)(3 - x^5)^2 - 10x^4(4x - 2)^2(3 - x^5)
\]

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you apply the product rule to more complex functions?
2. What is the chain rule, and why is it important in this context?
3. How do higher-order derivatives of this function behave?
4. Can this function's derivative be simplified further?
5. How would you approach integration for this kind of expression?

**Tip**: When applying the product rule, be mindful of simplifying each term after differentiation to avoid mistakes later on.

Differentiate the function: y = (4x - 2)^2 (3 - x^5)^2

Learn how to differentiate the function y = (4x - 2)^2 (3 - x^5)^2 using the product rule and chain rule. This detailed solution walks through each step of the differentiation process, including applying the product rule and using the chain rule. Suitable for college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation