https://file.aiquickdraw.com/mathbot/file/1729129868640hmr5vriw.jpg

To solve this problem, we need to find \( \frac{dx}{dy} \) using implicit differentiation for the given equation:

\[
3y^8 + x^4y^4 + 5yx^5 = 4y + 5
\]

### Step-by-step process:

1. **Differentiate both sides with respect to \( y \)**. Since we're looking for \( \frac{dx}{dy} \), we treat \( x \) as a function of \( y \) (i.e., implicit differentiation), so when differentiating terms with \( x \), we apply the chain rule.

2. **Differentiate the left-hand side**:
    - The derivative of \( 3y^8 \) with respect to \( y \) is:
      \[
      3 \cdot 8y^7 = 24y^7
      \]
    - For \( x^4y^4 \), use the product rule:
      \[
      \frac{d}{dy}(x^4y^4) = 4x^3 \frac{dx}{dy} \cdot y^4 + x^4 \cdot 4y^3 = 4x^3 y^4 \frac{dx}{dy} + 4x^4 y^3
      \]
    - For \( 5yx^5 \), again apply the product rule:
      \[
      \frac{d}{dy}(5yx^5) = 5 \cdot \frac{d}{dy}(yx^5) = 5 \left( x^5 + y \cdot 5x^4 \frac{dx}{dy} \right) = 5x^5 + 25yx^4 \frac{dx}{dy}
      \]

3. **Differentiate the right-hand side**:
    - The derivative of \( 4y + 5 \) with respect to \( y \) is:
      \[
      4
      \]
    - The constant 5 differentiates to 0.

Now, combine all the terms:

\[
24y^7 + 4x^3 y^4 \frac{dx}{dy} + 4x^4 y^3 + 5x^5 + 25yx^4 \frac{dx}{dy} = 4
\]

4. **Group the \( \frac{dx}{dy} \) terms** on one side:
\[
4x^3 y^4 \frac{dx}{dy} + 25yx^4 \frac{dx}{dy} = 4 - 24y^7 - 4x^4 y^3 - 5x^5
\]

Factor out \( \frac{dx}{dy} \) on the left-hand side:

\[
\frac{dx}{dy} \left( 4x^3 y^4 + 25yx^4 \right) = 4 - 24y^7 - 4x^4 y^3 - 5x^5
\]

5. **Solve for \( \frac{dx}{dy} \)**:
\[
\frac{dx}{dy} = \frac{4 - 24y^7 - 4x^4 y^3 - 5x^5}{4x^3 y^4 + 25yx^4}
\]

This is the expression for \( \frac{dx}{dy} \).

---

Would you like more details on any specific part of the process, or do you have questions?

### Related Questions:
1. What is implicit differentiation, and why is it used here?
2. How do you apply the product rule in differentiation?
3. Why is the chain rule necessary when differentiating terms with \( x \) in implicit differentiation?
4. How do you interpret the final result of implicit differentiation?
5. Can you perform similar implicit differentiation for more complex equations?

### Tip:
When using implicit differentiation, always apply the chain rule when differentiating terms involving both \( x \) and \( y \), especially if one variable is a function of the other.

Find dx/dy using implicit differentiation, where 3y^8 + x^4y^4 + 5yx^5 = 4y + 5.

Learn how to find dx/dy using implicit differentiation for the equation 3y^8 + x^4y^4 + 5yx^5 = 4y + 5. This problem covers advanced calculus concepts, including the product rule and chain rule. Suitable for high school students in grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation