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Implicit differentiation is a powerful technique used when dealing with functions in which one variable is not isolated. Instead of requiring the function to be explicit, this method allows us to differentiate implicitly by treating one variable as a function of another.
For example, in logarithmic differentiation, we often apply implicit differentiation to differentiate both sides of an equation.

Here's how it works:

• Differentiation is performed with respect to one variable (usually denoted as \( x \)).
• Every time you differentiate a term involving a function of \( x \), it is treated as a function, and implicit differentiation is applied.
• This technique is especially useful when differentiating complicated composite functions.

In our example, implicit differentiation is applied after using properties of logarithms to simplify the logarithmic expression before differentiating it with respect to \( x \). By doing so, we can find the derivative without explicitly solving the function for \( x \).

The derivative of logarithmic functions allows us to handle expressions involving logarithms effectively. Logarithmic differentiation is especially helpful when functions are products, quotients, or powers, as seen in complicated functions.
The key rule for differentiation of a natural log (\( \ln \)) function is:

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

where \( u \) is a differentiable function of \( x \) and \( u' \) is its derivative.

This rule essentially transforms the process of differentiating complex products or quotients by converting them into sums or differences that are easier to handle.
In the exercise, the application of this rule is pivotal in differentiating the expanded logarithmic expressions derived from the original function.
By taking the derivative of these logs, we convert multiplicative and divisive relationships in the function to additive and subtractive, streamlining the differentiation process.

The properties of logarithms, like the Power Rule and the Quotient Rule, are essential tools in simplifying complex logarithmic expressions. These properties allow us to rewrite logarithmic functions in more manageable forms, which is valuable during differentiation.

Here's a quick recap of the important properties of logarithms:

• Quotient Rule: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \)
• Power Rule: \( \ln(a^b) = b\ln(a) \)

These rules enable us to break down a fraction or a power in the argument of the log function into sums or differences.
In the provided exercise, applying these properties allowed us to express the natural log of the given function in terms of the natural logs of its components.
This transformation simplifies the differentiation process by removing the complexity introduced by exponential or fractional terms. Understanding and applying these properties is crucial for making logarithmic differentiation less burdensome and more straightforward.