https://file.aiquickdraw.com/mathbot/file/17266738678001t1w4tmc.jpg

To find \(\frac{dy}{dx}\) from the given equation, we need to differentiate implicitly. The equation is:

\[
54x^6 + 10x^{36}y + y^8 = 65
\]

### Step-by-step implicit differentiation:

1. **Differentiate each term with respect to \(x\)**:
   - The derivative of \(54x^6\) is:
     \[
     \frac{d}{dx}(54x^6) = 54 \cdot 6x^5 = 324x^5
     \]
   
   - The derivative of \(10x^{36}y\) requires the product rule, since it involves both \(x\) and \(y\):
     \[
     \frac{d}{dx}(10x^{36}y) = 10\left(\frac{d}{dx}(x^{36}) \cdot y + x^{36} \cdot \frac{dy}{dx}\right)
     \]
     \[
     = 10(36x^{35}y + x^{36}\frac{dy}{dx})
     \]
   
   - The derivative of \(y^8\) using the chain rule is:
     \[
     \frac{d}{dx}(y^8) = 8y^7\frac{dy}{dx}
     \]
   
   - The derivative of the constant \(65\) is 0.

2. **Combine the derivatives**:
   After differentiating each term, we get the equation:
   \[
   324x^5 + 10(36x^{35}y + x^{36}\frac{dy}{dx}) + 8y^7\frac{dy}{dx} = 0
   \]

3. **Solve for \(\frac{dy}{dx}\)**:
   Collect the terms involving \(\frac{dy}{dx}\):
   \[
   324x^5 + 360x^{35}y + 10x^{36}\frac{dy}{dx} + 8y^7\frac{dy}{dx} = 0
   \]
   Move the terms without \(\frac{dy}{dx}\) to the right-hand side:
   \[
   10x^{36}\frac{dy}{dx} + 8y^7\frac{dy}{dx} = -324x^5 - 360x^{35}y
   \]
   Factor out \(\frac{dy}{dx}\):
   \[
   \frac{dy}{dx}(10x^{36} + 8y^7) = -324x^5 - 360x^{35}y
   \]
   Finally, solve for \(\frac{dy}{dx}\):
   \[
   \frac{dy}{dx} = \frac{-324x^5 - 360x^{35}y}{10x^{36} + 8y^7}
   \]

This is the implicit derivative of \(y\) with respect to \(x\).

Do you have any questions or would you like further details?

### Related Questions:
1. How does implicit differentiation work for equations involving both \(x\) and \(y\)?
2. Can you apply the product rule to more complicated expressions?
3. How do we handle chain rule in higher-order derivatives?
4. What is the importance of implicit differentiation in calculus?
5. How would you interpret the derivative geometrically in this context?

### Tip:
Always simplify terms involving both \(x\) and \(y\) carefully, especially when using the product and chain rules together.

Given the equation below, find dy/dx. The equation is 54x^6 + 10x^36y + y^8 = 65

Explore how to apply implicit differentiation to solve for dy/dx in the equation 54x^6 + 10x^36y + y^8 = 65. This guide covers key calculus concepts including derivatives, the product rule, and the chain rule.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation