Problem 20 - Calculus - AP Edition

Question

Find \(d y / d x\). $$y=\ln (\ln (\ln x))$$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = \frac{1}{x \ln x \ln(\ln x)} \)

Step 1: Identify the Function and Its Composition

The given function is a composition of logarithmic functions: \(y = \ln(\ln(\ln x))\). To differentiate this, we need to use the chain rule multiple times.

Step 2: Apply the Chain Rule for the First Derivative

Apply the chain rule to differentiate the outer ln function: If \(z = \ln(\ln x)\), then \(y = \ln(z)\). The derivative is \(\frac{dy}{dz} = \frac{1}{z}\).

Step 3: Differentiate the Middle Layer

Now differentiate \(z = \ln(\ln x)\). Let \(u = \ln x\), then \(z = \ln(u)\) and \(\frac{dz}{du} = \frac{1}{u}\).

Step 4: Differentiate the Innermost Layer

Since \(u = \ln x\), differentiate to get \(\frac{du}{dx} = \frac{1}{x}\).

Step 5: Combine the Derivatives Using the Chain Rule

The chain rule gives us: \[ \frac{dy}{dx} = \frac{dy}{dz} \cdot \frac{dz}{du} \cdot \frac{du}{dx} = \frac{1}{\ln(\ln x)} \cdot \frac{1}{\ln x} \cdot \frac{1}{x} = \frac{1}{x \ln x \ln(\ln x)}. \]

Step 6: Present the Final Derivative

Therefore, the derivative of the function \(y = \ln(\ln(\ln x))\) with respect to \(x\) is \(\frac{dy}{dx} = \frac{1}{x \ln x \ln(\ln x)}\).

Key Concepts

Understanding Derivatives

The concept of the derivative is central to calculus. It measures how a function changes as its input changes. In other words, it tells us the rate at which one quantity changes with respect to another. In the context of our problem, we aim to find how the function \(y = \ln(\ln(\ln x))\) changes with respect to \(x\).
To do this, we use different rules that help us understand derivatives for different types of functions.

• Basic rule: The derivative of a simple term \(x^n\) is \(nx^{n-1}\).
• Logarithms: Use specific rules for functions involving logs, as in our exercise.
• Chain Rule: Useful for functions that are compositions of one or more functions.

Applying these rules appropriately simplifies the problem, allowing us to break down complex derivatives step by step.

Exploring Logarithmic Functions

Logarithmic functions are inverse of exponential functions and appear frequently in calculus.For a logarithmic function like \(y = \ln(x)\):

• The base is the number \(e\) (approximately 2.718).
• The derivative of \(\ln(x)\) with respect to \(x\) is \(\frac{1}{x}\).

In our problem, we are dealing with a series of nested logarithmic functions \(\ln(\ln(\ln x))\), which makes it a more complex function to differentiate. Each layer of \(\ln\) affects how we calculate the derivative. This is where recognizing the pattern and utilizing the chain rule effectively is key.
When dealing with nested or repeated logarithmic expressions, separate them step by step. Identify each logarithmic portion, apply their derivative rule, and then combine using the chain rule.

Composition of Functions and the Chain Rule

The chain rule is a fundamental tool in calculus for dealing with the composition of functions. This rule is what permits us to find the derivative of a function that is made up of several other functions.In our given function \(y = \ln(\ln(\ln x))\) the following compositions occur:

• The outermost function is \(y = \ln(z)\)
• The middle function is \(z = \ln(u)\)
• The innermost function is \(u = \ln(x)\)

When differentiating each layer, you start from the outermost to the innermost function. This stepwise differentiation using the chain rule looks like this:

• The derivative of \(y = \ln(z)\) is \(\frac{1}{z}\).
• The derivative of \(z = \ln(u)\) is \(\frac{1}{u}\).
• The derivative of \(u = \ln(x)\) is \(\frac{1}{x}\).

Finally, multiply these individual derivatives: \(\frac{1}{z}\times\frac{1}{u}\times\frac{1}{x}\), simplifying to give us the derivative \(\frac{1}{x \ln(x) \ln(\ln(x))}\).
This showcases how the chain rule simplifies the differentiation process for composite functions.