Question

For the following exercises, find the derivative \(d y / d x\). (You can use a calculator to plot the function and the derivative to confirm that it is correct.) $$ \text { [T] } y=\ln (\ln x) $$

Short answer

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

Step-by-step solution

Identify the Outer and Inner Functions

In the expression \( y = \ln (\ln x) \), we have two layers of functions. The outer function is \( f(u) = \ln u \) where \( u = \ln x \), and the inner function is \( g(x) = \ln x \).

Differentiate the Outer Function

Using the chain rule, start by finding the derivative of the outer function with respect to its inner variable \( u \). The derivative of \( f(u) = \ln u \) with respect to \( u \) is \( \frac{d}{du}[\ln u] = \frac{1}{u} \).

Differentiate the Inner Function

Next, find the derivative of the inner function \( g(x) = \ln x \) with respect to \( x \). The derivative is \( \frac{d}{dx}[\ln x] = \frac{1}{x} \).

Apply the Chain Rule

According to the chain rule, the derivative of \( y = \ln (\ln x) \) with respect to \( x \) is the product of the derivative of the outer function with respect to \( u \) and the derivative of \( u \) with respect to \( x \). This gives us: \( \frac{dy}{dx} = \frac{1}{\ln x} \cdot \frac{1}{x} \).

Simplify the Expression

Simplify the expression for the derivative to obtain the final answer: \( \frac{dy}{dx} = \frac{1}{x \ln x} \).

Key concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. A composite function is essentially a function within another function, like a nested set of equations.

• To apply the chain rule, identify the outer and inner functions in your equation.
• In our exercise, where we differentiate \( y = \ln(\ln x) \), the outer function is \( f(u) = \ln u \) and the inner function is \( g(x) = \ln x \).
• The chain rule states that to find the derivative of a composite function, you take the derivative of the outer function with respect to the inner function, then multiply it by the derivative of the inner function with respect to \( x \).

For example, the outer function's derivative is \( \frac{d}{du}[\ln u] = \frac{1}{u} \), and for the inner function, \( \frac{d}{dx}[\ln x] = \frac{1}{x} \). Therefore, by multiplying these, we find the derivative of \( y = \ln(\ln x) \) with respect to \( x \). This application of the chain rule allows us to correctly navigate the layers of the composite functions.

Logarithmic Functions

Logarithmic functions are crucial in various fields, particularly in differentiation. They are the inverse operations of exponential functions.

• The natural logarithm \( \ln x \) has a derivative of \( \frac{1}{x} \).
• In our exercise, we encounter a doubly nested logarithm \( \ln(\ln x) \), which adds complexity as there are two layers to differentiate.
• The key to differentiating such a function is recognizing the need for the chain rule.

When differentiating \( y = \ln(\ln x) \), begin with the innermost function and work outward. Each layer of the logarithmic function can be individually handled, starting with the inner \( \ln x \), whose derivative is simple and leads to the eventual differentiation of the whole expression. Understanding logarithmic properties is essential when they are combined with other functions.

Differentiation Steps

Differentiation involves a sequence of steps that simplifies the finding of derivatives, especially when dealing with composite functions. To solve the exercise, let's break it down:1. **Identify the Components**: Start by identifying the outer and inner functions, as seen with \( y = \ln(\ln x) \).2. **Differentiate the Outer Function**: Differentiate the outer part, \( \ln u \), treating the inner function (\( u \)) as a variable. Here it results in \( \frac{1}{u} \).3. **Differentiate the Inner Function**: Next, find the derivative of the inner function. For our example, this is \( \ln x \), leading to \( \frac{1}{x} \).4. **Apply the Chain Rule**: Multiply these derivatives together as per the chain rule guidance. So, \( \frac{dy}{dx} = \frac{1}{\ln x} \times \frac{1}{x} \).5. **Simplify**: Simplify the resulting expression to find \( \frac{dy}{dx} = \frac{1}{x \ln x} \).These steps ensure a structured approach to tackling derivatives, especially when logarithmic and composite functions are involved. Simplicity and logical order are the keys to success in differentiation.