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The cornerstone of calculus, the derivative measures how a function changes as its input changes. It's akin to determining the rate of change or the slope of a function at any given point. Using the limit definition, we say that the derivative of a function at a point x, denoted as \(g'(x)\), is the limit of the difference quotient as the increment approaches zero: \[ g'(x) = \lim_{{h \to 0}} \frac{{g(x+h) - g(x)}}{{h}} \]This formula encapsulates how tiny changes in x, represented by h, affect the function's output. The provided exercise uses this definition to find the derivative of a rational function. By evaluating this limit, we gain insight into the instantaneous rate of change of \(g(x)\) at any point x.

Simplifying complex fractions is critical in calculus, particularly when handling derivatives via the limit definition. A complex fraction involves a fraction in the numerator, the denominator, or both. Simplifying such fractions requires careful manipulation of both the numerator and the denominator to ultimately ensure the terms are in a form that facilitates further mathematical operations.In our exercise, simplification of the complex fraction involves combining \(g(x+h)\) and \(g(x)\) into a single fraction, multiplying out, and finally, eliminating terms with the factor h in both the numerator and the denominator. This step is essential to avoid undefined expressions and to reach a form where the limit as h approaches zero can be evaluated.

Evaluating limits is a fundamental process in calculus used to find the value that a function approaches as the input approaches a certain value. This concept is pivotal when calculating derivatives, because we are interested in the behavior of the function as our change in x, denoted as h, becomes infinitesimally small.In the exercise, we reach the stage where we need to evaluate \( \lim_{{h \to 0}} \) by substituting the value of zero for h. This step is straightforward when our expression is well-simplified, as no indeterminate forms such as \(\frac{0}{0}\) exist. The process requires careful algebraic manipulation to resolve into a determinate form from which we can deduce the limit, and thus, the derivative of the function.

Differentiation is the action of computing a derivative. The derivative describes the way a function changes at any point and allows us to understand dynamics such as velocity, acceleration, and growth rates in various fields. Differentiation can be approached in several ways, including the limit definition as practiced in our exercise, or through rules such as the power rule, product rule, quotient rule, and chain rule for more complex functions.In the context of the exercise, differentiation of the function \(g(x)\) using the limit definition provides a thorough understanding of the fundamental theorem of calculus, and an appreciation for the delicate process through which we ascertain the behavior of functions. Through the differentiation process, we conclude that the rate of change of \(g(x)\) with respect to x is zero, symbolizing a horizontal tangent line or a constant function.