Question

\(\mathrm{T} / \mathrm{F}:\) The Quotient Rule states that \(\frac{d}{d x}\left(\frac{x^{2}}{\sin x}\right)=\frac{\cos x}{2 x}\).

Short answer

False. The derivative using the Quotient Rule is \( \frac{2x \sin x - x^2 \cos x}{\sin^2 x} \), not \( \frac{\cos x}{2x} \).

Step-by-step solution

Determine Components of the Quotient

The Quotient Rule applies to functions in the form \( \frac{u}{v} \). For the given function \( \frac{x^{2}}{\sin x} \), we identify \( u = x^2 \) and \( v = \sin x \).

Calculate Derivatives of u and v

Find the derivative of \( u = x^2 \), which is \( u' = 2x \). Next, find the derivative of \( v = \sin x \), which is \( v' = \cos x \).

Apply the Quotient Rule Formula

The Quotient Rule states: \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2} \). Substitute \( u = x^2 \), \( u' = 2x \), \( v = \sin x \), and \( v' = \cos x \) into the formula: \[ \frac{d}{dx}\left(\frac{x^2}{\sin x}\right) = \frac{(2x)(\sin x) - (x^2)(\cos x)}{(\sin x)^2} \].

Simplify the Expression

From the above expression, simplify to get: \[ \frac{2x \sin x - x^2 \cos x}{\sin^2 x} \].

Evaluate the Given Expression

The statement in the exercise claims the derivative is \( \frac{\cos x}{2x} \), but our calculated result is \( \frac{2x \sin x - x^2 \cos x}{\sin^2 x} \). These two expressions are different.

Key concepts

Derivative Calculation

Calculating derivatives is one of the foundational skills in calculus. When we talk about derivatives, we are referring to the rate at which a function is changing at any given point. Imagine a car's speedometer; the number it shows is the derivative of the car's position over time, indicating how fast the car is going at each moment.

In mathematical terms, for a function \( f(x) \), the derivative \( f'(x) \) gives us this information about change. To calculate a derivative, one often begins by identifying the function whose derivative is required and applying the appropriate rule or formula for differentiation.

• The Power Rule, useful when dealing with powers of \( x \), states: \( \frac{d}{dx}[x^n] = nx^{n-1} \).
• The Chain Rule, beneficial when functions are composed with one another, is used to find the derivative of a composite function.
• The Quotient Rule is essential when one function is divided by another, given as \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2} \).

Understanding these rules, especially the Quotient Rule, is pivotal when working with problems that involve division of functions, like in the exercise provided. Adequately applying these rules ensures accurate calculation of derivatives.

Trigonometric Functions

Trigonometric functions are fundamental elements of mathematics that arise in several real-life contexts, especially in describing wave patterns and cyclic phenomena such as sound or light waves.

In calculus, the trigonometric functions like sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)) are not only pivotal in calculations but also are subject to differentiation and integration. These functions provide one of the key examples of periodic phenomena and their derivatives themselves have specific patterns that are used extensively in calculus.


Basic Derivatives of Trigonometric Functions:

• The derivative of \( \sin x \) is \( \cos x \).
• The derivative of \( \cos x \) is \(-\sin x \).
• The derivative of \( \tan x \) is \( \sec^2 x \).

When working with trigonometric functions, it is vital to become familiar with these derivatives. They come into play with problems involving periodic behavior and when the coefficients fluctuate rhythmically. Calculating these derivatives becomes second nature with practice and is a critical skill in solving complex calculus problems like the one in our example.

Differentiation Techniques

Differentiation techniques are methods developed to make finding derivatives easier, allowing us to tackle a wide range of functions. Differentiation involves applying mathematical operations to find a function's derivative, indicating the slope or gradient of the function at any point.

For complex functions, using basic rules might not suffice, which is why several differentiation techniques are valuable. For the function in our exercise, \( \frac{x^2}{\sin x} \), the use of the Quotient Rule is a perfect example of applying a specialized technique to address the challenge of deriving a ratio of functions.

Key Differentiation Techniques:

• Quotient Rule: Used when two functions are divided by each other.
• Product Rule: Applied when two functions are multiplied together. It states \( (uv)' = u'v + uv' \).
• Chain Rule: Utilized for the derivative of composite functions. It requires knowing the functions' compositions and their respective derivatives.

Each of these techniques serves its unique purpose and often they are used together in various combinations to simplify and solve complex calculus challenges. Mastery of these differentiation techniques is essential for anyone studying calculus, as they unlock the ability to differentiate not just basic functions but any function encountered.