Problem 5 - Calculus Early Transcendentals

Question

Find \(f^{\prime}(x)\) $$ f(x)=\frac{5-\cos x}{5+\sin x} $$

Step-by-Step Solution

Answer

The derivative is \( f^{\prime}(x) = \frac{5\sin x + 1 - 5\cos x}{(5 + \sin x)^2} \).

Step 1: Identify the Quotient Rule

To find the derivative of a function of the form \( \frac{u(x)}{v(x)} \), we use the quotient rule. The formula is given by: \( \left( \frac{u}{v} \right)^{\prime} = \frac{u^{\prime}v - uv^{\prime}}{v^2} \). In this problem, \( u(x) = 5 - \cos x \) and \( v(x) = 5 + \sin x \).

Step 2: Differentiate the Numerator

Find the derivative of the numerator \( u(x) = 5 - \cos x \). The derivative of a constant is zero, and the derivative of \( -\cos x \) is \( \sin x \). Therefore, \( u^{\prime}(x) = \sin x \).

Step 3: Differentiate the Denominator

Find the derivative of the denominator \( v(x) = 5 + \sin x \). Similarly, the derivative of \( \sin x \) is \( \cos x \), so \( v^{\prime}(x) = \cos x \).

Step 4: Apply the Quotient Rule

Substitute \( u(x), v(x), u^{\prime}(x), \) and \( v^{\prime}(x) \) into the quotient rule: \[ f^{\prime}(x) = \frac{(\sin x)(5 + \sin x) - (5 - \cos x)(\cos x)}{(5 + \sin x)^2} \].

Step 5: Simplify the Expression

Expand the numerator: \( \sin x (5 + \sin x) = 5\sin x + \sin^2 x \) and \( (5 - \cos x)\cos x = 5\cos x - \cos^2 x \). Substitute back: \[ f^{\prime}(x) = \frac{5\sin x + \sin^2 x - 5\cos x + \cos^2 x}{(5 + \sin x)^2} \].

Step 6: Further Simplify

Combine like terms in the numerator: \[ f^{\prime}(x) = \frac{5\sin x + \sin^2 x + \cos^2 x - 5\cos x}{(5 + \sin x)^2} \]. Recognize that \( \sin^2 x + \cos^2 x = 1 \), leading to: \[ f^{\prime}(x) = \frac{5\sin x + 1 - 5\cos x}{(5 + \sin x)^2} \].

Step 7: Present the Final Answer

The derivative of the function \( f(x) = \frac{5 - \cos x}{5 + \sin x} \) is \( f^{\prime}(x) = \frac{5\sin x + 1 - 5\cos x}{(5 + \sin x)^2} \).

Key Concepts

Understanding Derivatives with the Quotient Rule

The Quotient Rule is a method for finding the derivative of a function that is a quotient of two other functions. Think of it like dividing two parts and wanting to know how their rates of change compare. Specifically, if you have a function in the form of \( \frac{u(x)}{v(x)} \), the rule states that the derivative \( \left( \frac{u}{v} \right)^{\prime} \) is given by:

• Find the derivatives of the top (numerator) and bottom (denominator) parts separately.
• Apply the formula: \( \frac{u^{\prime}v - uv^{\prime}}{v^2} \).

Here, for the function \( f(x) = \frac{5 - \cos x}{5 + \sin x} \), the top is \( 5 - \cos x \) and the bottom is \( 5 + \sin x \). You need to find each of their derivatives first before plugging them into the formula. Always remember to square the denominator in the final derivative formula.

Trigonometric Functions in Derivatives

Trigonometric functions like sine and cosine play a significant role when differentiating. They frequently appear and have straightforward derivatives:

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

In our example, \( 5 - \cos x \) simplifies so that only \( -\cos x \) needs differentiation. Similarly, in \( 5 + \sin x \), only \( \sin x \) changes. This is because the derivative of a constant like 5 is zero.
These rules allow for seamless differentiation, making managing expressions much easier. Always check the function components individually when trigonometric aspects come into play.

Simplifying the Derivative Expression

Once you've applied the quotient rule and found a derivative, a significant step follows: simplification. Simplifying makes expressions cleaner and easier to understand. Here's how to simplify effectively:

• First, distribute any multiplied terms across their parentheses.
• Next, combine like terms. For instance, \( 5\sin x + \cos^2 x \) in the expression.
• If you encounter identities, such as \( \sin^2 x + \cos^2 x = 1 \), use them to rewrite terms more simply.

In our problem, the expression became much clearer when \( \sin^2 x + \cos^2 x \) turned into 1. Such simplification steps are crucial for ensuring the final answer is as concise as possible, leading to easier interpretation and application in further studies or problem-solving scenarios.