The graph of y =x^{ln x} has one horizontal tangent line. Find an equation for it.

Verified step by step guidance

1. Step 1: Recognize that a horizontal tangent line occurs where the derivative of the function is zero. Therefore, we need to find the derivative of the function y = x^{\ln x}.
2. Step 2: To differentiate y = x^{\ln x}, use logarithmic differentiation. Start by taking the natural logarithm of both sides: \ln y = \ln(x^{\ln x}) = \ln x \cdot \ln x.
3. Step 3: Differentiate both sides with respect to x. The left side becomes \frac{1}{y} \cdot \frac{dy}{dx} using the chain rule, and the right side becomes \frac{d}{dx}(\ln x \cdot \ln x) using the product rule.
4. Step 4: Apply the product rule to differentiate \ln x \cdot \ln x: \frac{d}{dx}(\ln x \cdot \ln x) = \ln x \cdot \frac{1}{x} + \ln x \cdot \frac{1}{x} = 2 \cdot \frac{(\ln x)}{x}.
5. Step 5: Substitute back to find \frac{dy}{dx}: \frac{1}{y} \cdot \frac{dy}{dx} = 2 \cdot \frac{(\ln x)}{x}. Solve for \frac{dy}{dx} and set it to zero to find the x-value where the tangent is horizontal.