Problem 16 - Thomas Calculus

Question

In Exercises \(7-38,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=(\ln x)^{3} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the function form

We must find the derivative of the function \( y = (\ln x)^3 \). Notice that this is a composite function where we have an outer function \( g(u) = u^3 \) and an inner function \( u = \ln x \).

Step 2: Apply the chain rule

The chain rule is used when differentiating a composite function. According to the chain rule, if \( y = g(u) \) and \( u = h(x) \), then \( \frac{dy}{dx} = \frac{dg}{du} \cdot \frac{du}{dx} \). Here, \( g'(u) = 3u^2 \) and \( u = \ln x \).

Step 3: Differentiate the outer function

Using the chain rule, differentiate the outer function \( g(u) = u^3 \) with respect to \( u \). The derivative of \( u^3 \) is \( 3u^2 \), so we have \( g'(u) = 3(\ln x)^2 \).

Step 4: Differentiate the inner function

Now, differentiate the inner function \( u = \ln x \) with respect to \( x \). The derivative of \( \ln x \) is \( \frac{1}{x} \).

Step 5: Combine results with the chain rule

Substitute the derivatives we found into the chain rule formula: \( \frac{dy}{dx} = 3(\ln x)^2 \frac{1}{x} \). Simplify the expression to \( \frac{dy}{dx} = \frac{3(\ln x)^2}{x} \).

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus for finding the derivative of a composite function. It allows us to differentiate more complex expressions where one function is nested within another. For example, when we see a function like \( y = (\ln x)^3 \), we recognize it as a composite function. This is because it has an inner function \( u = \ln x \) and an outer function \( g(u) = u^3 \).
To apply the chain rule correctly, follow these steps:

• Differentiate the outer function with respect to the inner variable \( u \). In our example, the derivative of \( u^3 \) is \( 3u^2 \).
• Differentiate the inner function with respect to \( x \). Here, the derivative of \( \ln x \) is \( \frac{1}{x} \).
• Multiply these derivatives together to get the overall derivative \( \frac{dy}{dx} = \frac{dg}{du} \cdot \frac{du}{dx} \).

This process allows us to handle derivatives of functions that are not easily broken down into simpler parts without the notion of composite functions.

Derivative of Logarithmic Functions

Understanding the derivative of logarithmic functions is vital for calculus. It gives us tools to deal with growth patterns that appear in nature or economics. Specifically, the derivative of the natural logarithm function \( \ln x \) is quite simple—it is \( \frac{1}{x} \). This result reflects the rate of change of the logarithm as \( x \) changes. Since logarithms describe scaling relationships, their derivatives are widespread in applications like physics and statistics.
In our example, the function \( y = (\ln x)^3 \) involves the natural logarithm \( \ln x \) as the inner function. By differentiating \( \ln x \) first to get \( \frac{1}{x} \), we prepare to apply the chain rule for the composite structure of the entire function. Remember, knowing basic derivatives like that of \( \ln x \) makes dealing with more complex expressions manageable.

Composite Functions

Composite functions are essentially functions within functions, which makes them intriguing yet complex mathematical objects. Consider the function \( y = (\ln x)^3 \). Here, the outer function \( g(u) = u^3 \) is applied to the inner function \( u = \ln x \). The key to understanding composite functions is recognizing these layers. Once identified, we can use strategies like the chain rule to calculate derivatives efficiently.
Composite functions often appear when real-world problems involve multiple stages of process transformation or chaining behaviors. Mathematical modeling in engineering, computer science, and economics frequently uses such functions to depict multidimensional phenomena. Mastery of calculus, including techniques for differentiation like the chain rule, enhances your ability to explore these rich and complex scenarios.