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

Answer

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Answer

$\boxed{y' = \frac{\ln(y) - \frac{y}{x}}{\ln(x) - \frac{x}{y}}}$

Steps

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