Problem 23 - Thomas Calculus

Question

In Exercises \(5-36,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\ln (\ln x) $$

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dx} = \frac{1}{x \ln x} \).

Step 1: Identify the Function

The function given is \( y = \ln(\ln x) \) which is a composition of functions. It is the natural logarithm of another natural logarithm function.

Step 2: Use the Chain Rule

To find the derivative, we need to apply the chain rule. Recall that the chain rule states that \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \). Here, \( f(u) = \ln u \) and \( g(x) = \ln x \).

Step 3: Differentiate the Outer Function

Differentiate the outer function \( \ln u \) with respect to \( u \). The derivative is \( \frac{d}{du}\ln u = \frac{1}{u} \). Here, \( u = \ln x \).

Step 4: Differentiate the Inner Function

Differentiate the inner function \( \ln x \) with respect to \( x \). The derivative is \( \frac{d}{dx}\ln x = \frac{1}{x} \).

Step 5: Apply the Chain Rule

Combine the results from Step 3 and Step 4 using the chain rule: \[ \frac{dy}{dx} = \frac{1}{\ln x} \cdot \frac{1}{x} = \frac{1}{x \ln x} \].

Step 6: Finalize the Derivative Expression

The derivative of \( y = \ln(\ln x) \) with respect to \( x \) is \( \frac{dy}{dx} = \frac{1}{x \ln x} \). This is the final expression for the derivative.

Key Concepts

Chain Rule in Calculus

The chain rule is an essential technique in calculus for finding derivatives of complex functions. When you deal with a function that is composed of multiple functions, the chain rule allows you to differentiate them step by step.

Let's say you have a composite function, such as the one in the exercise: \(y = \ln(\ln x)\). This expression involves two functions: \(g(x) = \ln x\) (the inner function) and \(f(u) = \ln u\) (the outer function). The chain rule helps us find the derivative by using these steps:

• Differentiate the outer function: Find \(f'(u)\), where \(u = \ln x\). The derivative here is \(f'(u) = \frac{1}{u}\).
• Differentiate the inner function: Find \(g'(x)\). The derivative is \(g'(x) = \frac{1}{x}\).
• Combine these derivatives: According to the chain rule, \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\), which results in \(\frac{1}{\ln x} \cdot \frac{1}{x} = \frac{1}{x \ln x}\).

Thus, the chain rule systematically breaks down the differentiation process, making it manageable.

Understanding Natural Logarithms

The natural logarithm, denoted as \(\ln\), is a special kind of logarithm with the base \(e\), where \(e \approx 2.71828\). It is widely used in various mathematical applications due to its unique properties.

Some important characteristics of the natural logarithm include:

• \(\ln(e) = 1\): Since it is the logarithm of \(e\) to the base \(e\).
• The derivative \(\frac{d}{dx} \ln x = \frac{1}{x}\): This simple derivative plays a crucial role in integration and differentiation.
• \(\ln(ab) = \ln a + \ln b\): A critical property when dealing with products.
• \(\ln(a^b) = b \ln a\): Useful for dealing with powers and exponents within logarithmic expressions.

In the exercise, \(\ln(\ln x)\) involves taking the natural logarithm of another logarithm, showcasing the versatile nature of \(\ln\) in combining and simplifying expressions.

Composition of Functions

The composition of functions is a process of combining two or more functions to form a new one. It is akin to a chain reaction where the output of one function becomes the input for another.

In mathematics, when dealing with composite functions like \(y = \ln(\ln x)\), the goal is to decompose and understand each function's role:

• The inner function \(g(x) = \ln x\): This transforms the initial input \(x\) into \(\ln x\).
• The outer function \(f(u) = \ln u\): Takes the result of the inner function and applies another transformation, leading to \(\ln(\ln x)\).

This idea of composing functions is pivotal, especially in the realm of calculus, where the orderly progression through each function's effect makes complex problems manageable.

Recognizing and breaking down composite functions ensures that the differentiation process using rules like the chain rule is both straightforward and precise.