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In calculus, derivatives are essential as they express how a function changes. The derivative of a function at a point gives us the slope of the tangent line to the function at that point. This slope describes how steep the graph of the function is at that particular location. For the function in the exercise, \[ f(x) = x + \frac{1}{x} \], calculating the derivative involves applying the power rule and the chain rule. We find the derivative to be \[ f'(x) = 1 - \frac{1}{x^2} \], representing how the function's slope changes with respect to \( x \).Understanding derivatives helps us predict how the function behaves locally and provides insights into its instantaneous rate of change. This insight is crucial for solving this exercise, as we use the derivative to find the tangent line approximation.

The tangent line approximation is a powerful tool in calculus, often referred to as the linearization of a function. Essentially, this concept involves using the tangent line to approximate a function at a certain point. The general formula for this is:\[ L(x) = f(a) + f'(a)(x-a) \].In our example, \( f(x) = x + \frac{1}{x} \), with \( a = 1 \), the linearization becomes fairly straightforward. We first find \( f(1) = 2 \) and \( f'(1) = 0 \). This tells us that the tangent line is horizontal at \( x = 1 \), yielding \( L(x) = 2 \). When the derivative at a point is 0, the tangent line is horizontal, perfectly flat, indicating no instantaneous change at that point. Studying tangent line approximation helps in estimating values of functions that are otherwise complex to compute, especially when close to \( x = a \).

Function evaluation at a given point involves substituting the value of the variable into the function and simplifying the result. It's a basic yet vital step in approximation methods like linearization. In this exercise, evaluating the function \( f(x) = x + \frac{1}{x} \) at \( x = a = 1 \) yields \( f(1) = 2 \). Function evaluation provides us with the exact value of the function at a specific point, ensuring accuracy in subsequent calculations. This value is crucial for identifying where the tangent line intersects the graph of the function at the chosen point, \( x = a \). It's the cornerstone of the linearization formula, showing how we move from abstract function expressions to tangible numbers easily interpreted in real-world contexts.