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.

Key Concepts

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is a fundamental concept in calculus used to find slopes of tangent lines. For a function to have a horizontal tangent line, its derivative must equal zero at that point.

Critical Points

Critical points occur where the derivative of a function is either zero or undefined. These points are essential for identifying local maxima, minima, and points of inflection. In the context of the given function, finding critical points will help determine where the horizontal tangent line exists.

Exponential and Logarithmic Functions

The function y = x ln x involves both polynomial and logarithmic components. Understanding the properties of logarithmic functions, such as their growth rates and behavior, is crucial for analyzing the function's graph and finding its derivative. This knowledge aids in solving for the horizontal tangent line effectively.