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Understanding exponential functions is crucial when dealing with growth patterns, be it in nature, finances, or even populations. An exponential function is defined as a function of the form
\(f(x) = a \times e^{kx}\), where \(e\) is the base of the natural logarithm, approximately equal to 2.71828. The numbers \(a\) and \(k\) are constants, with \(a\) representing the initial amount and \(k\) being the rate at which the exponential function grows or decays.

In the context of derivatives, when evaluating \(e^{x}\), it simplifies beautifully because the rate of change of \(e^{x}\) is also \(e^{x}\). This relation holds true for any exponent of \(e\), that is, \(e^{kx}\) will have a derivative of \(ke^{kx}\). Thus, for the given exercise where \(f(x)=2e^{x}\), the exponential form simplifies our understanding and calculation of the derivative.

In calculus, derivative rules are akin to a map for navigating the terrain of functions. They provide us with formulas and methods to find the rate of change, or derivatives, for various kinds of functions.
There are several key rules, such as the power rule, the product rule, the quotient rule, and the chain rule. For exponential functions, where we have \(e^{x}\), we use a specific rule that states that the derivative of \(e^{x}\) is \(e^{x}\) itself.

In the exercise, this rule is applied to find \(f'(x)\) of an exponential function, resulting in an almost identical expression but with the constant coefficient preserved. That's why our derived function is \(f'(x) = 2 \times e^{x}\). Rules like these streamline the process of differentiation by providing a clear set of instructions, which, when followed, lead to the correct derivative of the function in question.

Function estimation is a technique to deduce a function's output values without computing them with complete accuracy, and it often comes into play when dealing with irrational numbers like \(e\). When we find the derivative \(f'(x) = 2 \times e^{x}\), estimating \(f'(1)\) means we're finding the slope of the tangent line to the curve at \(x=1\).

The value of \(e\), which is irrational, is often rounded for simplicity. However, in many cases—including the exercise—we retain the exact value \(e\) for precision when we describe the derivative. By substituting \(x=1\) back into the derivative, we estimated the value of \(f'(1)\) to be \(2e\), an accurate representation of the function's rate of change at that specific point. Always remember, estimation doesn't necessarily compromise accuracy—it often simplifies the expression while retaining the exact value.