Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \underline{\phantom{xxx}} \text { find } D_{x} y . $$ $$ y=\frac{\sin x}{x} $$

Step-by-Step Solution

Answer

The derivative is \( D_x y = \frac{x \cos x - \sin x}{x^2} \).

Step 1: Understand the Derivative Definition

The problem asks us to find the derivative of the function \( y = \frac{\sin x}{x} \) with respect to \( x \). This involves applying rules of differentiation to each component of the function.

Step 2: Apply the Quotient Rule

Since \( y \) is a quotient of two functions, we use the quotient rule: if \( u = \sin x \) and \( v = x \), then the derivative \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2} \). Here, \( u' = \cos x \) and \( v' = 1 \).

Step 3: Calculate Components of the Quotient Rule

Substitute the derivatives into the quotient rule formula: \[ D_x y = \frac{x \cdot \cos x - \sin x \cdot 1}{x^2} = \frac{x \cos x - \sin x}{x^2} \]

Step 4: Simplify the Result

This calculation simplifies to the final result: \[ D_x y = \frac{x \cos x - \sin x}{x^2} \] This is the derivative of \( y = \frac{\sin x}{x} \) with respect to \( x \).

Key Concepts

Quotient Rule

The Quotient Rule is a fundamental part of differentiation techniques used in calculus. When you have a function that is a division of two functions, such as \( y = \frac{u}{v} \), the Quotient Rule helps you find the derivative. The formula for the Quotient Rule is:

• \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2} \)

This means that to differentiate a quotient, you:

• Multiply the derivative of the numerator (\( u' \)) by the denominator (\( v \))
• Subtract the product of the numerator (\( u \)) and the derivative of the denominator (\( v' \))
• Divide the whole expression by the square of the denominator \( (v^2) \)

The Quotient Rule is especially handy when dealing with functions that contain other complex functions, streamlining the differentiation process.

Trigonometric Functions

Trigonometric functions, such as \( \sin x \), \( \cos x \), and \( \tan x \), are some of the most common functions you’ll encounter in calculus. These functions describe the relationships between the angles and sides of triangles and are foundational in modeling periodic phenomena like waves.
When differentiating trigonometric functions, it is vital to remember these basic derivatives:

• \( \frac{d}{dx} \sin x = \cos x \)
• \( \frac{d}{dx} \cos x = -\sin x \)
• \( \frac{d}{dx} \tan x = \sec^2 x \)

In the given exercise, \( y = \frac{\sin x}{x} \), knowing that \( \frac{d}{dx} \sin x = \cos x \) is crucial, because it directly influences how you apply the Quotient Rule. These derivatives form the building blocks for more complicated derivative calculations.

Differentiation Techniques

Differentiation techniques are strategies and rules in calculus used to find the derivative of functions. Understanding and utilizing these techniques is essential for solving various problems in mathematics and engineering.
The main techniques include:

• Power Rule: For any function \( f(x) = x^n \), \( \frac{d}{dx} x^n = nx^{n-1} \).
• Product Rule: Used when differentiating the product of two functions, \( (uv)' = u'v + uv' \).
• Quotient Rule: Used for ratios as discussed above.
• Chain Rule: Useful for composite functions, \( \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \).

By mastering these techniques, you can tackle complex differentiation problems efficiently. Each method has specific applications based on the structure of the function you are dealing with, making them indispensable tools in calculus.