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To find \(\frac{dy}{dx}\) using implicit differentiation from the equation:

\[
x^5 + y^5 = 15xy
\]

### Step-by-Step Solution:

1. Differentiate both sides of the equation with respect to \(x\), treating \(y\) as a function of \(x\). Remember to apply the **product rule** and **chain rule** where necessary.

   \[
   \frac{d}{dx}(x^5) + \frac{d}{dx}(y^5) = \frac{d}{dx}(15xy)
   \]

2. Differentiate each term:
   - The derivative of \(x^5\) is:
     \[
     5x^4
     \]
   - The derivative of \(y^5\) (using the chain rule since \(y\) is a function of \(x\)) is:
     \[
     5y^4 \frac{dy}{dx}
     \]
   - For the right-hand side, use the **product rule** for \(15xy\):
     \[
     \frac{d}{dx}(15xy) = 15 \left( x \frac{dy}{dx} + y \right)
     \]

3. Now, substitute these into the differentiated equation:
   \[
   5x^4 + 5y^4 \frac{dy}{dx} = 15(x \frac{dy}{dx} + y)
   \]

4. Rearrange the terms to collect all \(\frac{dy}{dx}\) terms on one side:
   \[
   5y^4 \frac{dy}{dx} - 15x \frac{dy}{dx} = 15y - 5x^4
   \]

5. Factor out \(\frac{dy}{dx}\):
   \[
   \frac{dy}{dx}(5y^4 - 15x) = 15y - 5x^4
   \]

6. Solve for \(\frac{dy}{dx}\):
   \[
   \frac{dy}{dx} = \frac{15y - 5x^4}{5y^4 - 15x}
   \]

7. Simplify the expression if possible:
   \[
   \frac{dy}{dx} = \frac{3y - x^4}{y^4 - 3x}
   \]

This is the derivative \(\frac{dy}{dx}\) using implicit differentiation.

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Would you like further details or any clarifications?

### Related Questions:
1. How do you apply the product rule in differentiation?
2. What is the chain rule, and why is it important in implicit differentiation?
3. Can you find the second derivative \(\frac{d^2y}{dx^2}\) from this equation?
4. How would the process change if \(y\) was a constant?
5. How does implicit differentiation differ from explicit differentiation?

### Tip:
Always carefully apply the product and chain rules when differentiating terms involving both \(x\) and \(y\), as this helps to avoid common mistakes in implicit differentiation.

Use implicit differentiation to find dy/dx. Given equation: x^5 + y^5 = 15xy

Learn how to find dy/dx using implicit differentiation for the equation x^5 + y^5 = 15xy. The solution covers key calculus concepts, including the product rule and chain rule. Suitable for students in Grade 12 or early college-level calculus.

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