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Differentiating logarithmic functions can be straightforward with the right approach.Logarithmic differentiation is a valuable method, especially when dealing with complex functions such as quotients or products. To differentiate a natural log function like \( \ln u \), where \( u \) is a function of \( x \), you apply the formula:

• \( \frac{d}{dx} \ln u = \frac{u'}{u} \)

This formula signifies that the derivative of \( \ln u \) is the derivative of \( u \), noted as \( u' \), divided by \( u \) itself.

For the given function \( \ln \frac{1+e^{x}}{1-e^{x}} \), you must first identify \( u \) as the fraction \( \frac{1+e^{x}}{1-e^{x}} \). Differentiating this correctly requires further steps, applying other rules like the Chain Rule and Quotient Rule in tandem to tackle the underlying complexity.

The Chain Rule is a powerful tool in calculus to differentiate composite functions.A composite function is one in which input to one function is another function.If \( f(x) = g(h(x)) \), then to find \( f'(x) \), you utilize the Chain Rule:

• \( f'(x) = g'(h(x)) \cdot h'(x) \)

This rule essentially states that the derivative of the outer function \( g \) at \( h(x) \) must be multiplied by the derivative of \( h(x) \).

In our exercise, after identifying \( y = \ln(\frac{1+e^x}{1-e^x}) \), we apply the Chain Rule for differentiation by substituting the derivative of the inside function (the log argument) using the Quotient Rule. Once we've calculated \( y' \) using the Quotient Rule,
we multiply this by the derivative of the outer natural logarithm function according to the Chain Rule.

The Quotient Rule is indispensable when differentiating ratios of functions.Whenever two functions \( u(x) \) and \( v(x) \) are involved such that they form a quotient, \( \frac{u}{v} \),
this rule is handy for finding the derivative.

• The Quotient Rule is given by \( \left(\frac{u}{v}\right)' = \frac{vu' - uv'}{v^2} \).

In our given function, \( u = 1+e^x \) and \( v = 1-e^x \). After calculating \( u' = e^x \) and \( v' = -e^x \),
the rule helps in concluding that the derivative \( y' = \frac{2e^x}{(1-e^x)^2} \).
The key is to be careful in applying the rule by maintaining the order of subtraction precisely and not forgetting to square the denominator.