Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=\pi x $$

Step-by-Step Solution

Answer

The derivative \( D_{x} y \) is \( \pi \).

Step 1: Identify the Components

The given function is a simple linear function: \( y = \pi x \). Here, \( \pi \) is a constant coefficient, and \( x \) is the variable.

Step 2: Apply the Constant Multiplication Rule

The differentiation rule for constant multiplication states that if you have a function \( y = a \, u(x) \), where \( a \) is a constant and \( u(x) \) is a differentiable function of \( x \), then the derivative \( D_{x}(y) = a \cdot D_{x}(u(x)) \). In our case, \( a = \pi \) and \( u(x) = x \).

Step 3: Differentiate the Variable

Differentiate \( x \) with respect to \( x \). The derivative of \( x \) is 1 (since the derivative of \( x^n \) with respect to \( x \) is \( nx^{n-1} \), and here \( n=1 \)).

Step 4: Combine the Results

According to the constant multiplication rule, multiply the constant \( \pi \) by the derivative of \( x \), which is 1. Therefore, \( D_{x}(y) = \pi \times 1 = \pi \).

Key Concepts

Constant Multiplication Rule

Understanding differentiation can sometimes feel complex, but the constant multiplication rule simplifies things significantly. It helps us find the derivative when a function involves a constant multiplied by a variable.

Here's how it works:

• Consider a function of the form \( y = a \cdot u(x) \), where \( a \) is a constant and \( u(x) \) is a differentiable function of \( x \).
• The constant multiplication rule allows us to "ignore" the constant initially and focus on finding the derivative of the function \( u(x) \).
• After finding the derivative of \( u(x) \), multiply the result by the constant \( a \).

This rule is incredibly useful because it simplifies calculations, letting us focus on differentiating the variable part directly before incorporating the constant factor.

Derivative

The concept of a derivative is foundational in calculus. It represents the rate at which a function is changing at any given point.

When we talk about derivatives, we're often interested in how a quantity changes as something else changes. Specifically for a function \( y = f(x) \), the derivative \( D_{x}(y) \) of \( y \) with respect to \( x \) tells us:

• How steep the curve is at a particular point, which means the slope of the tangent line.
• The direction of change; a positive derivative indicates increasing values, while a negative derivative indicates decreasing values.

For basic functions like \( y = x \), finding the derivative is straightforward. The derivative of \( x \), with respect to itself, is 1. This directly stems from the power rule, which states \( nx^{n-1} \). When \( n = 1 \), we get \( 1 \times x^{0} = 1 \). This simplicity empowers us to tackle more complex derivatives by understanding these basic ones first.

Linear Function

A linear function is one of the simplest forms of a function in mathematics. It can be expressed in the form \( y = mx + b \), where:

• \( m \) is the slope of the line, showing how steep the line is, or how much \( y \) changes for a unit change in \( x \).
• \( b \) is the y-intercept, indicating where the line crosses the y-axis.

In our specific case, the function \( y = \pi x \) is linear because it fits the template \( y = mx + b \), with \( m = \pi \) and \( b = 0 \).

The beauty of linear functions lies in their simplicity. They graph as straight lines, and their derivatives are constant \( m \), consistent with their constant slope. When you differentiate linear functions such as \( \pi x \), the derivative directly gives us the constant \( \pi \), reinforcing how straightforward working with such functions can be.