Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the Product Rule to show that \(D_{x}[f(x)]^{2}=\) \(2 \cdot f(x) \cdot D_{x} f(x)\)

Step-by-Step Solution

Answer

By applying the product rule to \([f(x)]^2 = f(x) \times f(x)\), we derive \(D_{x}[f(x)]^2 = 2f(x)D_{x}f(x)\).

Step 1: Understanding the Product Rule

The Product Rule in calculus states that if you have two functions, say \( u(x) \) and \( v(x) \), the derivative of their product is \( D_{x}[u(x)v(x)] = u(x)D_{x}v(x) + v(x)D_{x}u(x) \). We'll apply this to \([f(x)]^2 = f(x) imes f(x)\).

Step 2: Apply the Product Rule

Set \( u(x) = f(x) \) and \( v(x) = f(x) \). Now using the Product Rule, \( D_{x}[f(x)f(x)] = f(x)D_{x}f(x) + f(x)D_{x}f(x) \).

Step 3: Simplify the Expression

Combine like terms in the previous step: \( f(x)D_{x}f(x) + f(x)D_{x}f(x) = 2f(x)D_{x}f(x) \).

Step 4: Finalize the Result

We have shown that \( D_{x}[f(x)]^2 = 2f(x)D_{x}f(x) \) as required by applying the Product Rule and combining like terms.

Key Concepts

Understanding Differentiation

Differentiation is a fundamental concept in calculus that revolves around finding the rate at which a function changes.
It allows us to determine the slope of the tangent line to the curve at any given point.
This is crucial in many real-world applications, such as physics and engineering, where understanding how quantities change over time or space is vital.In our exercise, we discuss the differentiation using the Product Rule, which is a specialized method for dealing with products of functions. For instance, when we have a function like

• \( [f(x)]^2 \), which is essentially \( f(x) imes f(x) \),

the Product Rule helps us find the derivative of this expression effectively.By practicing differentiation, you not only hone your problem-solving skills but also gain insights into the behavior of different functions.
It's a stepping stone to learning more complex topics in calculus.

The Concept of Derivative

A derivative represents how a function changes as its input changes.
It is a core idea in calculus that all students must become familiar with.
In essence, the derivative measures the instantaneous rate of change, which can be thought of as the slope of the curve at a specific point.In the context of our problem, taking the derivative of

• \( [f(x)]^2 \)

requires us to calculate how this function changes as \( x \) changes.Using the Product Rule, we explore how derivatives behave in products of functions.
The derivative of a product \( u(x)v(x) \) is given by

• \( u(x)D_{x}v(x) + v(x)D_{x}u(x) \) .

This allows us to derive complex expressions in an organized manner.
Once you grasp this concept, applying derivatives to various functions will become more intuitive and manageable.

Effective Calculus Problem Solving with Product Rule

Calculus problem solving often involves using systematic methods to break down complex expressions.
The Product Rule is one such tool that simplifies the derivation process for functions expressed as products.In our specific problem, applying the Product Rule helps us find the derivative of

• \( [f(x)]^2 \) by treating it as \( f(x) imes f(x) \).

Each function \( f(x) \) is treated separately to find its derivative and then combined according to the rule.Here's a brief recap of how you should approach similar problems:

• Identify the functions involved in the product.
• Assign them as \( u(x) \) and \( v(x) \).
• Apply the Product Rule formula: \( u(x)D_{x}v(x) + v(x)D_{x}u(x) \).
• Simplify the resulting expression.

By doing so, you can tackle calculus problems with confidence.
Remember, practice is key! Applying these methods repeatedly will sharpen your calculus skills and make problem solving more straightforward.