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A derivative represents the rate at which a function's value changes as its input changes. Essentially, it provides an instantaneous rate of change. Suppose you have a function, like the one in the exercise, where \( f(x) = 2e^x \). The act of finding the derivative, \( f'(x) \), involves applying differentiation rules to determine how the function's output changes with a small increase in \( x \).

For an exponential function such as \( e^x \), the derivative is simple because \( \frac{d}{dx}(e^x) = e^x \). So it becomes \( f'(x) = 2e^x \). This tells us how sharply the function rises or falls at any given point. The derivative is a crucial part of calculus, used widely in physics, engineering, and economics to model and predict changing systems.

The slope of a tangent line at a specific point gives us a snapshot of how steeply the function is increasing or decreasing at that location. In the context of the problem, after identifying the derivative function \( f'(x) = 2e^x \), we use this to find the slope at a particular point, like \( x = 0 \).

By plugging \( x = 0 \) into the derivative, we calculated \( f'(0) = 2e^0 = 2 \). This means that right at the point \((0,2)\), the tangent to the curve rises by 2 units for every 1 unit it moves horizontally. This slope of 2 forms the backbone of constructing the tangent's equation using point-slope form, showing us how detailed and vital derivatives are in predictive calculations.

Function differentiation is the action of calculating the derivative. It involves using rules or formulas to find the derivative of a function. In most basic cases, this involves powerful techniques like product and chain rules, especially for more complex or combined functions.

For the function in the exercise, \( f(x) = 2e^x \), its differentiation is straightforward because you multiply the coefficients and the exponential terms directly. Therefore, you obtain \( f'(x) = 2e^x \) almost as a direct replacement using the known rule \( \frac{d}{dx}(e^x) = e^x \).

Differentiation is foundational in calculus, allowing mathematicians and scientists to explore and solve real-world problems involving change and continuous variation.