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Logarithmic functions have significant importance in mathematics, especially when dealing with complex equations involving exponential growth or decay. Understanding how to derive these functions is essential for numerous applications in calculus and real-world scenarios. The derivative of the natural logarithm function, denoted as \(\ln(x)\), is particularly simple yet powerful. When we differentiate \(\ln(x)\), we obtain \(\frac{1}{x}\).

This derivative illustrates how the logarithmic function's slope changes at any given point along its curve. A steeper slope signifies a larger derivative, indicating a more significant rate of change of the function's value. For example, for \(x > 1\), the function \(\ln(x)\) grows slowly, and thus its derivative decreases, emphasizing the function's diminishing rate of change as x increases.

The concept of instantaneous rate of change is a pivotal idea in calculus, and it refers to the slope of the tangent line to a function at any given point. This is closely associated with the derivative of a function. If we envision a particle moving along the path defined by a function, the instantaneous rate of change at any point would be equivalent to the particle's velocity at that specific moment.

For logarithmic functions like \(\ln(x)\), the instantaneous rate of change at a point x reveals how swiftly the value of the function is increasing or decreasing as x changes. In the context of the exercise, the value \(f'(1) = 1\) represents the instantaneous rate of change of \(\ln(x)\) when \(x=1\), indicating that at this point, the function's value changes unit for unit with x.

Calculating the derivative of a function is a key skill in calculus that involves determining the exact rate at which a function's output changes relative to changes in its input. This procedure is known as differentiation. For our example with \(f(x)=\ln(x)\), differentiation tells us that the slope, or rate of change, of \(\ln(x)\) at any point is \(\frac{1}{x}\).

In practice, when given a specific change in x, denoted as \(dx\), we can calculate the corresponding change in the function's output, \(dy\), by employing the derivative. As shown in the solution, \(dy\) is obtained by multiplying the derivative \(f'(x)\) with the infinitesimal increment \(dx\). This method is highly effective in predicting small changes in the function's output and is fundamental for analyzing the behavior of functions across various fields like physics, economics, and engineering.