Problem 26 - Thomas Calculus

Question

In Exercises \(7-38,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\ln (\ln (\ln x)) $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Function Composition

The given function is a composition of three natural logarithm functions: \[ y = \ln(\ln(\ln x)) \]To find its derivative, we need to apply the chain rule multiple times.

Step 2: Apply the Chain Rule (First Layer)

Start by differentiating the outermost function. The outermost function is \(\ln(u)\) where \(u = \ln(\ln x)\). The derivative of \(\ln(u)\) is \(\frac{1}{u}\). Thus, \[ \frac{dy}{du} = \frac{1}{\ln(\ln x)} \]

Step 3: Apply the Chain Rule (Second Layer)

Now differentiate the middle function, which is again a natural logarithm, \(\ln(v)\) where \(v = \ln x\). The derivative of \(\ln(v)\) is \(\frac{1}{v}\). Thus,\[ \frac{du}{dv} = \frac{1}{\ln x} \]

Step 4: Apply the Chain Rule (Innermost Layer)

Finally, differentiate the innermost function, which is \(\ln x\). The derivative of this is \(\frac{1}{x}\). Thus,\[ \frac{dv}{dx} = \frac{1}{x} \]

Step 5: Combine Derivatives Using the Chain Rule

Using the chain rule, the overall derivative \(\frac{dy}{dx}\) is the product of the derivatives from each layer:\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dx} \] Substitute the values found in the previous steps:\[ \frac{dy}{dx} = \frac{1}{\ln(\ln x)} \cdot \frac{1}{\ln x} \cdot \frac{1}{x} \]Thus,\[ \frac{dy}{dx} = \frac{1}{x \ln x \ln(\ln x)} \]

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate compositions of functions. It is particularly useful when you have a function inside another function. Imagine peeling an onion; you handle it one layer at a time.
In practical terms, when you have a composite function like \[ y = f(g(x)) \] the derivative of \( y \) with respect to \( x \) is found by multiplying the derivative of the outer function, \( f \), by the derivative of the inner function, \( g \). Thus, the chain rule is written as \[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \].
When applying the chain rule multiple times, like in the case of \( y = \ln(\ln(\ln x)) \), you work with each logarithm step-by-step:

• Differentiate the outermost layer first.
• Work your way inward, differentiating each subsequent layer.
• Multiply the derivatives together to get the overall derivative.

This multi-layer derivation significantly simplifies the process of differentiating complex composite functions.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a special logarithmic function with the base \( e \), where \( e \approx 2.71828 \). It is widely used in calculus for its beautiful properties and symmetry. When you differentiate \( \ln(x) \), you obtain the result \[ \frac{d}{dx} \ln(x) = \frac{1}{x} \].
This differentiation is straightforward because the natural logarithm's derivative always simplifies to \( \frac{1}{x} \), making it easier to work through layers of functions.
In the context of a composition like \( \ln(\ln(\ln x)) \), each natural logarithm layer is differentiated using the simple rule of \( \ln(u) \), which yields \[ \frac{1}{u} \].
This means you repeatedly apply this principle at each stage of the composition until you reach the innermost function. This highlights the beauty of calculus in breaking down complex expressions.

Function Composition

Function composition is the process of applying one function to the results of another. It is denoted by the symbol \( \circ \), and for functions \( f \) and \( g \), it is expressed as \( (f \circ g)(x) = f(g(x)) \).
Composition allows for the creation of complex functions by chaining simpler functions together. When it comes to differentiating these nested functions, each layer acts as a gate that influences the derivative of the composite function.
Consider the function \( y = \ln(\ln(\ln x)) \):

• The outer function is \( \ln \), applied to \( \ln(\ln x) \).
• The middle function is \( \ln \), applied to \( \ln x \).
• The innermost function is just \( \ln x \).

By understanding the hierarchy and applying the chain rule sequentially, each composition layer can be effectively differentiated. Thus, despite the intimidating appearance of deeply nested functions, differentiation becomes manageable by breaking it down step by step.