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When working with exponential functions in calculus, understanding how to find their derivatives is essential. Evidently, the derivative represents the rate at which the function's value changes. Exponential functions, which can be generally expressed as \(a^{x}\), where \(a\) is a constant and \(x\) is the exponent, have unique properties.

For functions with a variable as the base and the exponent like \(x^{2x}\), deriving them involves using the concept of logarithmic differentiation. Initially, you would take the natural logarithm of both sides, apply the properties of logarithms to simplify, and then differentiate implicitly. This method is powerful because it transforms a potentially complicated exponential derivative into a more manageable form. Understanding this process is fundamental because exponential functions are inherent in various fields including biology, economics, and physics.

The chain rule is one of the most powerful tools in calculus for finding the derivative of composite functions. A composite function is a function composed of two or more functions, where the output of one function becomes the input of another. The rule states that if \(u=g(x)\) and \(y=f(u)\), then the derivative of \(y\) with respect to \(x\) is \(dy/dx = (dy/du) * (du/dx)\).

The chain rule allows calculation of the derivative of complex functions by breaking them down into simpler parts for which derivatives can be more easily calculated. This step-by-step approach makes it easier to solve derivative problems that might first appear intimidating. Especially in the case of finding horizontal tangent lines, where you're often dealing with compositions of functions, the chain rule can simplify the process of finding where the derivative is zero.

In calculus, finding the equation of a tangent line to a curve at a given point involves calculating the derivative of the function at that point. The derivative gives the slope of the tangent line, and knowing a point on the line allows us to find its equation using the point-slope form. Specifically, horizontal tangent lines occur where the derivative of the function is zero, signifying no change in the y-direction at those points. These are often critical points that may correspond to local minima or maxima.

In the given exercise, after calculating the derivative and setting it to zero, the resultant x-values are where the function has a horizontal tangent. You then use these x-values to find the corresponding y-values on the original function. Here, this approach led to the conclusion that the curve \(y = x^{2x}\) has a horizontal tangent not at two different heights, as might be typical, but rather coincidentally at the same y-value, yielding identical equations for both horizontal tangent lines: \(y=1\). This revelation highlights the importance of thoroughly analyzing the results when finding tangent lines.