Problem 8 - Calculus

Question

Find the derivative \(d y / d x\) in \(1-10\). $$ y=\ln (\ln x) $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Function Type

The function given is a composition of two natural logarithm functions, denoted as \(y = \ln(\ln x)\). We need to use the chain rule to differentiate this composition.

Step 2: Apply the Chain Rule

To apply the chain rule, let's identify the outer function and the inner function. The outer function is \(u = \ln v\) with \(v = \ln x\). We need to compute \(\frac{dy}{du}\) and \(\frac{du}{dx}\).

Step 3: Differentiate Outer Function

For the outer function \(u = \ln v\), the derivative is \(\frac{dy}{du} = \frac{1}{v}\).

Step 4: Differentiate Inner Function

For the inner function \(v = \ln x\), the derivative is \(\frac{du}{dx} = \frac{1}{x}\).

Step 5: Combine Using Chain Rule

Using the chain rule, \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\). Substituting the derivatives: \[ \frac{dy}{dx} = \frac{1}{\ln x} \cdot \frac{1}{x} = \frac{1}{x \ln x} \].

Key Concepts

Derivative

Calculus introduces us to the concept of a derivative, which essentially represents the rate of change of a function with respect to one of its variables. The derivative is a foundational tool in calculus used to find the slope of a function at any point. This slope indicates how steep the function is at that point, akin to finding how fast we are climbing or descending if the function were a hill.In a more formal setting, if you have a function, say \( y = f(x) \), the derivative \( \frac{dy}{dx} \) expresses how small changes in \( x \) result in changes in \( y \). It's important to know:

• If the derivative is positive, the function is increasing at that section.
• If the derivative is negative, the function is decreasing there.
• A zero derivative indicates a flat spot - a peak, valley, or a pause.

Being able to find the derivative of complex functions, as in our exercise with nested logarithmic functions, broadens our understanding of how intricate relationships behave.

Chain Rule

The chain rule is a powerful technique in calculus utilized for finding the derivative of composite functions. Composite functions are functions where one function is encapsulated within another. This rule breaks down the differentiation process into simpler steps. The basic idea of the chain rule can be captured as follows: if you have a function \( y = f(g(x)) \), then the derivative \( \frac{dy}{dx} \) is the product of the derivative of the outer function \( f \) evaluated at the inner function \( g(x) \), multiplied by the derivative of the inner function \( g \). In formula form, this is: \[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \] For the exercise, the function \( y = \ln(\ln x) \) involves a logarithm within another logarithm, making it a perfect candidate for applying the chain rule:

• The outer function is \( u = \ln v \), and its derivative is \( \frac{dy}{du} = \frac{1}{v} \).
• The inner function is \( v = \ln x \), with a simple derivative \( \frac{du}{dx} = \frac{1}{x} \).

By multiplying these derivatives together, the chain rule gives us the final derivative \( \frac{dy}{dx} = \frac{1}{x \ln x} \). It might seem complex, but each step breaks down into manageable parts.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a special logarithm in mathematics where the base is the mathematical constant \( e \), roughly equal to 2.71828. This form of logarithm is omnipresent in mathematics due to its powerful properties and natural emergence in continuous growth models. In terms of calculus, the derivative of the natural logarithm function is particularly straightforward: \[ \frac{d}{dx}(\ln x) = \frac{1}{x} \] This simplicity is why the natural logarithm often shows up as an intermediate step when differentiating more complex functions, just like in our given exercise where we differentiate \( \ln(\ln x) \).Key properties of the natural logarithm include:

• It's undefined for non-positive values of \( x \). \( \ln(x) \) only accepts positive real numbers as input.
• It converts multiplication into addition: \( \ln(ab) = \ln a + \ln b \).
• It's the inverse of the exponential function \( e^x \), meaning \( e^{\ln x} = x \).

The natural logarithm is an essential tool that simplifies many mathematical and real-world functions, embedding itself deeply into the structure of calculus.