In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
122. y = (ln x)^(ln x)

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Recognize that the function is of the form \(y = (\ln x)^{\ln x}\), which is a variable base raised to a variable exponent. This is a perfect candidate for logarithmic differentiation.

Take the natural logarithm of both sides to simplify the expression: \(\ln y = \ln \left( (\ln x)^{\ln x} \right)\).

Use the logarithm power rule to bring down the exponent: \(\ln y = (\ln x) \cdot \ln (\ln x)\).

Differentiate both sides with respect to \(x\). Remember to use implicit differentiation on the left side and the product rule on the right side: \(\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx} \left( (\ln x) \cdot \ln (\ln x) \right)\).

Apply the product rule to the right side: \(\frac{d}{dx} (\ln x) \cdot \ln (\ln x) + (\ln x) \cdot \frac{d}{dx} \ln (\ln x)\). Then find each derivative separately using the chain rule, and finally multiply both sides by \(y\) to solve for \(\frac{dy}{dx}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions where both the base and the exponent are variable expressions. By taking the natural logarithm of both sides, the function is transformed into a product or sum, simplifying differentiation using the chain and product rules.

Properties of Logarithms

Understanding logarithm properties, such as ln(a^b) = b ln(a), is essential for rewriting complex expressions. These properties allow the exponent to be brought down as a multiplier, making differentiation more straightforward when dealing with functions like (ln x)^(ln x).

Chain Rule and Product Rule

The chain rule is used to differentiate composite functions, while the product rule applies when differentiating products of functions. In logarithmic differentiation, after applying logarithms, these rules are often combined to find the derivative of the transformed expression.

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