Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-18, find \(D_{x} y\). $$ y=\frac{\sin x+\cos x}{\cos x} $$

Step-by-Step Solution

Answer

The derivative \( D_x y \) is \( \sec^2 x \).

Step 1: Identify the Equation Format

The given function is a rational function, where the numerator is \( \sin x + \cos x \) and the denominator is \( \cos x \). This means we can use the quotient rule to find the derivative.

Step 2: Apply the Quotient Rule

The quotient rule states that for a function \( y = \frac{u}{v} \), the derivative \( D_x y \) is \( \frac{v \cdot u' - u \cdot v'}{v^2} \), where \( u = \sin x + \cos x \) and \( v = \cos x \).

Step 3: Differentiate the Numerator and Denominator

Find the derivatives of \( u \) and \( v \). For \( u = \sin x + \cos x \), the derivative is \( u' = \cos x - \sin x \). For \( v = \cos x \), the derivative is \( v' = -\sin x \).

Step 4: Plug into the Quotient Rule Formula

Substitute \( u \), \( u' \), \( v \), and \( v' \) into the quotient rule formula: \[ D_x y = \frac{\cos x (\cos x - \sin x) - (\sin x + \cos x)(-\sin x)}{\cos^2 x} \].

Step 5: Simplify the Expression

Expand and simplify the expression: \[ \cos x (\cos x - \sin x) + (\sin x + \cos x) \sin x = \cos^2 x - \cos x \sin x + \sin^2 x + \cos x \sin x \]. This simplifies to \[ \cos^2 x + \sin^2 x \].

Step 6: Utilize Trigonometric Identity

Recognize that \( \cos^2 x + \sin^2 x = 1 \) from the Pythagorean identity.

Step 7: Final Expression of the Derivative

Thus, the derivative is \( D_x y = \frac{1}{\cos^2 x} \), which simplifies to \( \sec^2 x \).

Key Concepts

Quotient Rule

The quotient rule is a fundamental technique in calculus for finding the derivative of a quotient of two functions. When you have a function of the form \( y = \frac{u}{v} \), you need a way to calculate its derivative efficiently. The quotient rule provides this with its formula:

• \( D_x y = \frac{v \, u' - u \, v'}{v^2} \)

In this context:

• \( u \) is the numerator function.
• \( v \) is the denominator function.
• \( u' \) and \( v' \) are their respective derivatives.

By following the quotient rule, you avoid the complexity of differentiating the entire expression at once. It simplifies the process by breaking it down into manageable parts.

Trigonometric Functions

Trigonometric functions are fundamental in the study of calculus and appear frequently in problems involving geometrical shapes and periodic patterns. The primary functions include sine (\( \sin x \)), cosine (\( \cos x \)), and tangent (\( \tan x \)). In the exercise example, both sine and cosine functions are involved as:

• \( \sin x + \cos x \) forms the numerator.
• \( \cos x \) serves as the denominator of the rational function.

Understanding the derivatives of these functions can guide you through differentiation quickly:

• The derivative of \( \sin x \) is \( \cos x \).
• The derivative of \( \cos x \) is \( -\sin x \).

These basic derivatives are essential in applying the quotient rule in the original exercise.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which a function is changing at any point. This is crucial in calculus for analyzing and optimizing functions. When applying differentiation to complex expressions like a quotient of two functions, using rules such as the quotient rule can efficiently give you the derivative without directly computing limits.
In our example:

• Differentiation is applied to \( u = \sin x + \cos x \), resulting in \( u' = \cos x - \sin x \).
• The denominator \( v = \cos x \) is differentiated to yield \( v' = -\sin x \).

Be sure to follow each differentiation step accurately to avoid errors when inserting these into the quotient rule formula.

Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric identity often used in simplifying expressions involving sine and cosine. It states that:

• \( \cos^2 x + \sin^2 x = 1 \)

In the given exercise, after applying the quotient rule and differentiating, you obtain a complex expression. By recognizing parts of it as a Pythagorean identity, the expression \( \cos^2 x + \sin^2 x \) is simplified to \( 1 \). This simplification plays a vital role in determining the final expression of the derivative, which leads us to conclude:

• The derivative result is \( D_x y = \frac{1}{\cos^2 x} \) or \( \sec^2 x \).

The Pythagorean identity thereby reduces the complexity and aids in finding elegant solutions to trigonometric differentiation problems.