Question

Find $f^{\prime}(x)$ if $f(x)$ is the given expression. $$ (\ln x)^{\ln x} $$

   Find $f^{\prime}(x)$ if $f(x)$ is the given expression.
$$
(\ln x)^{\ln x}
$$

Calculus

Earl W. Swokowski 5th Edition

Chapter 7, Problem 30

Instant Answer

Solved by Expert Steven Clarke

03/15/2022

Step 1

Using the properties of logarithms, we can bring down the exponent to the front of the expression: \[ \ln y = \ln x \cdot \ln(\ln x) \] Show more…

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Find $f^{\prime}(x)$ if $f(x)$ is the given expression. $$ (\ln x)^{\ln x} $$

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Key Concepts

Exponential Function Differentiation

After rewriting the original function using the logarithmic identity, the function takes on an exponential form. Differentiating an exponential function involves multiplying the function by the derivative of its exponent, a process streamlined by the previous steps of using logarithms and the chain rule.

Logarithmic Differentiation

This technique is used when the function has variables in both the base and the exponent, as it allows one to take the natural logarithm of both sides of an equation to convert the exponentiation into multiplication. It simplifies the differentiation process by reducing the original problem to finding the derivative of a logarithmic expression.

Chain Rule

The chain rule is essential in differentiating composite functions. In this context, after rewriting the function in an exponential form, the chain rule helps in differentiating the outer exponential function and then multiplying by the derivative of the inner function.

Product Rule

When differentiating products of functions, such as when the inner function is a product of logarithmic terms, the product rule is used. It states that the derivative of a product is the derivative of the first function times the second function plus the first function times the derivative of the second function.

Properties of Logarithms

Logarithmic properties, such as the fact that the logarithm of an exponentiated expression can be expressed as a product, are instrumental in simplifying the function before differentiation. This involves rewriting the original function in a form that is easier to differentiate.

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