Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 47-58, express the indicated derivative in terms of the function \(F(x)\). Assume that \(F\) is differentiable. $$ D_{x}(F(2 x)) $$

Step-by-Step Solution

Answer

The derivative is \( 2F'(2x) \).

Step 1: Identify the Problem

The problem asks us to find the derivative of the function \( F(2x) \) with respect to \( x \). We assume that \( F \) is a differentiable function.

Step 2: Apply the Chain Rule

To differentiate \( F(2x) \), we use the chain rule, which states \( \frac{d}{dx}[F(g(x))] = F'(g(x)) \cdot g'(x) \). Here, \( g(x) = 2x \), so \( g'(x) = 2 \).

Step 3: Differentiate \( F(2x) \)

Using the chain rule, the derivative \( \frac{d}{dx}[F(2x)] \) becomes \( F'(2x) \cdot 2 \).

Step 4: Express the Final Derivative

The final expression for the derivative of \( F(2x) \) in terms of \( F \) is \( 2F'(2x) \).

Key Concepts

Understanding Derivatives

Derivatives are a fundamental concept in calculus and represent a way of measuring how a function changes as its input changes. Imagine driving a car: your speed at any precise moment is essentially the derivative of your position over time.
With functions, the derivative tells us how steep or flat the slope of the curve is at any point, informing us of how the function is increasing or decreasing. In mathematical terms, the derivative of a function \( f(x) \) at a point \( x \) is defined by the limit:

• \( f'(x) = \lim_{{h \to 0}} \frac{f(x + h) - f(x)}{h} \)

This formula might seem complex at first glance, but it beautifully captures the essence of instantaneous rate of change.
Learning how to differentiate functions and calculating derivatives becomes crucial when dealing with a variety of problems in calculus.

Function Differentiation

Function differentiation is the process of finding the derivative of a function. This critical process allows us to understand how a function behaves at various points.

• To differentiate a function, you often apply differentiation rules, such as the power rule, product rule, quotient rule, and chain rule.

By using these rules, you can break down even complex functions into understandable parts.
For example, when faced with a composite function like \( F(2x) \), as in our original exercise, we utilize the chain rule to tackle the differentiation.
Knowing when and how to apply these rules is vital, making practice with a range of function types important for mastering differentiation.

Calculus and the Chain Rule

Calculus is a branch of mathematics focused on change, whether that change involves rates or accumulations. Within calculus, one of the powerful tools is the chain rule, which is used for differentiating composite functions.
Given a function \( F(g(x)) \), where \( g(x) \) itself is a function of \( x \), the chain rule helps us find the derivative of this nested function efficiently. The rule states:

• \( \frac{d}{dx} [F(g(x))] = F'(g(x)) \cdot g'(x) \)

The chain rule can be thought of as a way to "chain together" derivatives of outside and inside functions.
In our example \( F(2x) \), \( g(x) = 2x \), and \( g'(x) = 2 \). Using the chain rule, the derivative becomes \( 2F'(2x) \), allowing us to easily see how changes in \( x \) affect \( F(2x) \).
Mastering the chain rule is important as it frequently appears in calculus problems.