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The chain rule is a fundamental technique in calculus for finding the derivative of composite functions. A composite function occurs when one function is nested inside another. For example, in our original problem, we deal with the expression \(\ln \left( |u| \right)\), where \(u\) itself is a function of \(x\), namely \(\frac{x}{x+1}\).
To apply the chain rule, follow these basic steps:

• Differentiate the outer function with respect to the inner function.
• Multiply this result by the derivative of the inner function with respect to \(x\).

This process allows us to effectively "chain" the derivatives together. For instance, if \(y = \ln |u|\), then the derivative \(\frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx}\). In our case, once we identify \(u = \frac{x}{x+1}\), we then find the derivative of \(u\) in the next step, demonstrating how chain rule and other differentiation techniques work together.

Logarithmic differentiation is a special technique especially useful for functions involving logarithms, products, or quotients powered to variable exponents. The advantage of using logarithmic differentiation is that it simplifies complex differentiation problems by using logarithm properties such as \(\ln(a^b) = b \cdot \ln(a)\).
In our exercise, we first simplified \(\ln \sqrt{| \frac{x}{x+1} |}\) as \(\frac{1}{2} \ln \left| \frac{x}{x+1} \right|\). This transforms the differentiation task from a situation involving a square root to a simpler problem.
The use of \(\frac{1}{2} \ln |u|\) allows us to separate the power from the inside of the logarithm, reducing potential errors and clarifying our steps. By simplifying the expression, all subsequent derivative operations become much more straightforward.

The quotient rule in calculus is specifically for differentiating expressions where one function is divided by another. In its standard form, if \(v = \frac{f(x)}{g(x)}\), then its derivative \(v'\) is given by:

• \(v' = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}\).

This rule allows us to differentiate complex rational functions by treating them in two separate parts: the numerator and the denominator. In our solution, the inner function \(u = \frac{x}{x+1}\) is differentiated using the quotient rule. Here, \(f(x) = x\) and \(g(x) = x+1\). The derivatives \(f'(x) = 1\) and \(g'(x) = 1\) lead us to compute \(\frac{1}{(x+1)^2}\) as the expression's derivative.
Using the quotient rule gives precision in dealing with the complexities of functions within fractions, ensuring each step maintains structural integrity of the problem.