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The equation provided is \( y = x^x + y^y \), and it asks to find the derivative \( \frac{dy}{dx} \) for \( x, y > 0 \).

To solve this, we'll differentiate both sides of the equation with respect to \( x \), using implicit differentiation since \( y \) is a function of \( x \). Let's proceed step-by-step:

1. Differentiate \( y = x^x + y^y \) with respect to \( x \):

   - For \( x^x \), we use logarithmic differentiation:
     \[
     \frac{d}{dx}(x^x) = x^x(\ln x + 1)
     \]
   - For \( y^y \), we apply implicit differentiation:
     \[
     \frac{d}{dx}(y^y) = y^y(\ln y + 1)\frac{dy}{dx}
     \]

2. Now, differentiate both sides:
   \[
   \frac{dy}{dx} = \frac{d}{dx}(x^x) + \frac{d}{dx}(y^y)
   \]
   Substituting the differentiated forms:
   \[
   \frac{dy}{dx} = x^x(\ln x + 1) + y^y(\ln y + 1)\frac{dy}{dx}
   \]

3. Rearranging the equation to solve for \( \frac{dy}{dx} \):
   \[
   \frac{dy}{dx} - y^y(\ln y + 1)\frac{dy}{dx} = x^x(\ln x + 1)
   \]
   Factor out \( \frac{dy}{dx} \):
   \[
   \frac{dy}{dx}\left(1 - y^y(\ln y + 1)\right) = x^x(\ln x + 1)
   \]
   Finally, solve for \( \frac{dy}{dx} \):
   \[
   \frac{dy}{dx} = \frac{x^x(\ln x + 1)}{1 - y^y(\ln y + 1)}
   \]

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

Would you like a detailed explanation of any step or further clarifications?

Here are some related questions for deeper understanding:
1. What is logarithmic differentiation, and why is it used for functions like \( x^x \)?
2. Can you explain how implicit differentiation is applied to functions where both sides involve the variable \( y \)?
3. How does the derivative of \( y^y \) change if \( y \) were independent of \( x \)?
4. What happens to the derivative expression when \( y = 1 \)?
5. How would this process change if the equation involved powers of other variables (e.g., \( y = x^z + y^z \))?

**Tip:** When differentiating powers involving the variable itself, logarithmic differentiation simplifies the process, allowing easier handling of such functions.

y = x^x + y^y  න් dy/dx සොයන්න.(x, y > 0)

This solution explores how to find the derivative dy/dx for the equation y = x^x + y^y using implicit differentiation and logarithmic differentiation techniques. It walks through each step, including the application of the chain rule and product rule, with detailed explanations.

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