To find the derivative of the equation $x^y = y^x$, we can proceed with implicit differentiation. Here's the step-by-step approach:

Step 1: Take the natural logarithm of both sides

First, take the natural logarithm of both sides to simplify the equation:

$\ln(x^y) = \ln(y^x)$

Using the logarithmic identity $\ln(a^b) = b \ln(a)$, this becomes:

$y \ln(x) = x \ln(y)$

Step 2: Differentiate implicitly

Now, differentiate both sides with respect to $x$, keeping in mind that $y$ is a function of $x$ (i.e., $y = y(x)$).

• Left side: Use the product rule to differentiate $y \ln(x)$.

$\frac{d}{dx} \left( y \ln(x) \right) = \frac{dy}{dx} \ln(x) + y \frac{1}{x}$

• Right side: Use the product rule to differentiate $x \ln(y)$.

$\frac{d}{dx} \left( x \ln(y) \right) = \ln(y) + x \frac{1}{y} \frac{dy}{dx}$

Step 3: Set up the equation

Now, equate the derivatives from both sides:

$\frac{dy}{dx} \ln(x) + \frac{y}{x} = \ln(y) + \frac{x}{y} \frac{dy}{dx}$

Step 4: Solve for $\frac{dy}{dx}$

Move the terms involving $\frac{dy}{dx}$ to one side:

$\frac{dy}{dx} \ln(x) - \frac{x}{y} \frac{dy}{dx} = \ln(y) - \frac{y}{x}$

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

$\frac{dy}{dx} \left( \ln(x) - \frac{x}{y} \right) = \ln(y) - \frac{y}{x}$

Now solve for $\frac{dy}{dx}$:

$\frac{dy}{dx} = \frac{\ln(y) - \frac{y}{x}}{\ln(x) - \frac{x}{y}}$

Final Answer:

The derivative of the equation $x^y = y^x$ with respect to $x$ is:

$\frac{dy}{dx} = \frac{\ln(y) - \frac{y}{x}}{\ln(x) - \frac{x}{y}}$

Let me know if you want further clarification or details on any part!

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Related Questions:

1. How can we apply implicit differentiation to equations involving both $x$ and $y$?
2. What are some special cases for the equation $x^y = y^x$ when $x = y$?
3. How would you solve for $y$ explicitly if possible?
4. What is the geometric interpretation of this equation and its derivative?
5. How does the derivative behave when $x$ and $y$ approach certain values?

Tip: When differentiating equations like $x^y = y^x$, taking the logarithm of both sides often simplifies the process significantly!