Problem 32 - Calculus for Biology and Medicine

Question

Differentiate the functions with respect to the independent variable. $$ f(x)=(\ln x)^{3} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{3(\ln x)^2}{x} \).

Step 1: Identify the Function Type

The function given is \( f(x) = (\ln x)^3 \). It is a composite function with an outer function \( u^3 \) and an inner function \( u = \ln x \). Differentiation will use the chain rule.

Step 2: Differentiate the Outer Function

First, treat \( \ln x \) as a single variable \( u \). The derivative of \( u^3 \) with respect to \( u \) is \( 3u^2 \).

Step 3: Differentiate the Inner Function

Next, find the derivative of \( u = \ln x \) with respect to \( x \). The derivative is \( \frac{1}{x} \).

Step 4: Apply the Chain Rule

According to the chain rule, the derivative of \( f(x) = (\ln x)^3 \) with respect to \( x \) is given by \( 3(\ln x)^2 \cdot \frac{1}{x} \).

Step 5: Simplify the Result

Multiply the terms to write the derivative in a simpler form: \( \frac{3(\ln x)^2}{x} \). Thus, the derivative \( f'(x) = \frac{3(\ln x)^2}{x} \).

Key Concepts

Composite Function

A composite function is a function that is made up of two or more functions. Think of it like nesting one function inside another. For instance, if you have a function represented as \( f(g(x)) \), you are essentially replacing the variable in \( f(x) \) with \( g(x) \). In the context of our exercise, the original function \( f(x) = (\ln x)^3 \) is a composite of the outer function \( u^3 \) and the inner function \( u = \ln x \).

• Outer Function: \( u^3 \) - This part evaluates the cubic component.
• Inner Function: \( \ln x \) - This function involves the natural logarithm of \( x \).

Understanding composite functions is fundamental in calculus as they frequently appear in various mathematical situations, providing a basis for differentiation using advanced rules.

Chain Rule

The chain rule is a powerful tool in calculus used for differentiating composite functions. It allows us to break down complex derivatives into manageable steps. The basic idea is to differentiate the outer function and then multiply it by the derivative of the inner function.
In our exercise, the chain rule demonstrates how to handle the derivative of \( f(x) = (\ln x)^3 \). Here's a step-by-step explanation:

• Differentiation of the outer function: Consider \( u = \ln x \) as a separate variable. The derivative of \( u^3 \) with respect to \( u \) is \( 3u^2 \).
• Differentiation of the inner function: The derivative of \( \ln x \) with respect to \( x \) is \( \frac{1}{x} \).
• Applying the chain rule: The derivative of the composite function is then \( 3(\ln x)^2 \cdot \frac{1}{x} \).
• Final simplified form: \( \frac{3(\ln x)^2}{x} \).

Using the chain rule effectively simplifies otherwise challenging problems, making difficult calculus tasks far more approachable.

Natural Logarithm

The natural logarithm, often expressed as \( \ln x \), is a logarithmic function with base \( e \), where \( e \) is approximately 2.718. This special kind of logarithm is prevalent in calculus because of its unique properties and the ease it provides in differentiation.
When dealing with derivatives involving logarithms, the function \( \ln x \) has a derivative that is especially simple: \( \frac{1}{x} \). This fits perfectly when using the chain rule, as illustrated in the given exercise. Here's why natural logarithms matter:

• Efficiency: The rule is straightforward—\( \text{If } \, f(x) = \ln x, \, f'(x) = \frac{1}{x} \). This simple derivative is powerful for simplifying expressions.
• Common in exponential functions: Functions related to growth and decay, such as interest rates and population models, often involve \( e \) and \( \ln x \).
• Helps in solving equations: Logarithms can transform multiplicative relationships into additive ones—useful for solving complex mathematical problems.

Understanding the natural logarithm and its properties helps in mastering calculus, especially when applied to differentiate and integrate functions involving \( e \).