Find $y^{\prime}$ if $x^{y}=y^{x}$. \[ y^{\prime}= \]

Step 1 ：\(\ln(x^{y}) = \ln(y^{x})\)

Step 2 ：\(y\ln(x) = x\ln(y)\)

Step 3 ：\(y'\ln(x) + \frac{y}{x} = \ln(y) + \frac{xy'}{y}\)

Step 4 ：\(y'(\ln(x) - \frac{x}{y}) = \ln(y) - \frac{y}{x}\)

Step 5 ：\(\boxed{y' = \frac{\ln(y) - \frac{y}{x}}{\ln(x) - \frac{x}{y}}}\)