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The chain rule is a fundamental technique in calculus used for finding the derivative of composite functions. In its essence, the chain rule provides a way to differentiate a function that is made up of one function inside another. For instance, if you have a function \(h(x) = g(f(x))\), the derivative \(h'(x)\) will be \(g'(f(x)) \times f'(x)\).

Here's why this is important in logarithmic differentiation. When taking the derivative of a logarithmic function that involves another function—as we see with \(\frac{d}{dx}\big(\text{ln}(f(x))\big)\)—the chain rule allows us to differentiate the outer function, which is the natural logarithm, and multiply it by the derivative of the inner function, \(f(x)\). This is precisely what we utilize when applying the derivative to \(\text{ln}(x)\) or \(\text{ln}(\text{cos}(x))\) in the given exercise. The chain rule ensures that every layer of the function is accounted for in the differentiation process.

Understanding the properties of the natural logarithm is crucial when simplifying expressions before differentiation. The natural logarithm, denoted as \(\text{ln}(x)\), has several key properties that make logarithmic differentiation handy. These properties include:

• The product rule: \(\text{ln}(xy) = \text{ln}(x) + \text{ln}(y)\).
• The quotient rule: \(\text{ln}\big(\frac{x}{y}\big) = \text{ln}(x) - \text{ln}(y)\).
• The power rule: \(\text{ln}(x^a) = a \text{ln}(x)\).

During the step-by-stepprocess, we visit the second step wherein we simplify the complex function using these properties. The original function \(f(x)\) includes a product, a quotient, and an exponent within its structure. By applying natural logarithm properties, the equation was significantly simplified and prepared for the application of the derivative. These properties allowed us to break down the composite logarithm into the sum and difference of several simpler logarithms, each of which is easier to differentiate.

When dealing with the derivative of logarithmic functions, there are straightforward rules to follow. For example, the derivative of the natural logarithm of x, denoted as \(\text{ln}(x)\), with respect to x is \(\frac{d}{dx}(\text{ln}(x)) = \frac{1}{x}\). Furthermore, the differentiation of more complex logarithmic functions often involves utilizing the chain rule, as discussed earlier.

In our exercise, step three is where these rules come into play. After simplifying the logarithmic expression in the previous step, we differentiate both sides of the equation. The derivatives of \(\text{ln}(x)\) and \(\text{ln}(\text{cos}(x))\) are found using the aforementioned rule and the chain rule, producing a term-by-term differentiation that leads to \(\frac{f'(x)}{f(x)}\). This relationship is crucial as it allows us to isolate \(f'(x)\) by multiplying both sides by \(f(x)\) in the subsequent step. The exercise exemplifies how the derivative of each logarithmic part is determined and then combined to find the overall derivative of the original function.