Problem 16 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-18, find \(D_{x} y\). $$ y=\frac{x \cos x+\sin x}{x^{2}+1} $$

Step-by-Step Solution

Answer

\(D_{x} y = \frac{-x^3 \sin x - 3x \sin x + 2 \cos x}{(x^2 + 1)^2}\)

Step 1: Identify Function Components

The given function is a quotient: \(y = \frac{u}{v}\), with \(u = x \cos x + \sin x\) and \(v = x^2 + 1\). We need to apply the quotient rule to differentiate this function.

Step 2: Differentiate the Numerator

Differentiate \(u = x \cos x + \sin x\). This involves using the product rule for \(x \cos x\) and the basic derivative of \(\sin x\). \(\frac{du}{dx} = \frac{d}{dx}(x \cos x) + \frac{d}{dx}(\sin x) \).\( = [x(-\sin x) + \cos x] + \cos x \).\( = -x \sin x + 2 \cos x \).

Step 3: Differentiate the Denominator

Find the derivative of the denominator \(v = x^2 + 1\). This is straightforward: \(\frac{dv}{dx} = \frac{d}{dx}(x^2) + \frac{d}{dx}(1) \).\(= 2x + 0 = 2x\).

Step 4: Apply the Quotient Rule

Use the quotient rule \(D\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}\). Substitute the derivatives and original functions: \(D_{x}y = \frac{(x^2 + 1)(-x \sin x + 2 \cos x) - (x \cos x + \sin x)(2x)}{(x^2 + 1)^2}\).

Step 5: Simplify the Expression

Expand and simplify: 1. Expand the terms:\((x^2 + 1)(-x \sin x + 2 \cos x) = -x^3 \sin x + 2x^2 \cos x - x \sin x + 2 \cos x\;(x \cos x + \sin x)(2x) = 2x^2 \cos x + 2x \sin x\).2. Simplify the numerator:\(-x^3 \sin x + 2x^2 \cos x - x \sin x + 2 \cos x - 2x^2 \cos x - 2x \sin x = -x^3 \sin x - 3x \sin x + 2 \cos x\).3. The entire expression:\(D_{x} y = \frac{-x^3 \sin x - 3x \sin x + 2 \cos x}{(x^2 + 1)^2}\).

Key Concepts

Understanding the Quotient Rule

In calculus differentiation, the quotient rule is a technique used when you need to differentiate a function that is the ratio of two differentiable functions. Specifically, if you have a function expressed as a quotient \( y = \frac{u}{v} \), where \( u \) and \( v \) are functions of \( x \), the quotient rule states that the derivative \( D_x \left( \frac{u}{v} \right) \) is given by:

• \( \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \)

This rule can be thought of as a method to properly handle the division of functions when taking derivatives, ensuring that both the numerator's and denominator's rates of change are accounted for.
Before applying the quotient rule, always identify \( u \) (numerator) and \( v \) (denominator) and calculate their respective derivatives, \( \frac{du}{dx} \) and \( \frac{dv}{dx} \).
Finally, substitute these derivatives and simplify the result, ensuring you carefully account for all algebraic operations.

Exploring the Product Rule

The product rule is another fundamental technique in calculus differentiation, essential when differentiating a product of two functions. For a function \( u(x) \) and \( v(x) \), the derivative of the product \( y = u \cdot v \) is given by:

• \( \frac{d}{dx}(u \cdot v) = u \cdot \frac{dv}{dx} + v \cdot \frac{du}{dx} \)

The product rule is crucial whenever you encounter expressions where two functions are multiplied together.
To apply the product rule, you need to individually differentiate each function in the product while simultaneously considering each function's contribution to the product's total rate of change.
In the given exercise, when differentiating \( x \cos x \), the product rule is crucial. Recognize \( u = x \) and \( v = \cos x \), then apply the formula.
Carefully performing these calculations ensures each component's effect on the overall differentiation is captured properly.

Introduction to Trigonometric Differentiation

Trigonometric differentiation involves finding derivatives of functions containing trigonometric terms like \( \sin x \) and \( \cos x \). These derivatives are fundamental and should be memorized for ease of use:

• Derivative of \( \sin x \) is \( \cos x \)
• Derivative of \( \cos x \) is \(-\sin x \)

Incorporating these into differentiation problems requires a good grasp of basic trigonometric identities and their derivatives.
The function \( u(x) = x \cos x + \sin x \) involves both standard and trigonometrically derived components. To differentiate it:

• Use the product rule on \( x \cos x \).
• Directly apply the derivative for \( \sin x \).

Trigonometric differentiation is incredibly useful and frequently appears in calculus, making understanding these principles key to mastering more complex problems.