Problem 32 - Calculus Single Variable

Question

Use the Quotient Rule to compute the derivative of the given expression with respect to \(x .\) $$ x / \sin (x) $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Functions for the Quotient Rule

Recognize the expression as a quotient where the numerator is \( u = x \) and the denominator is \( v = \sin(x) \). These are required for applying the quotient rule for derivatives.

Step 2: Recall the Quotient Rule Formula

The quotient rule is given by \( \frac{d}{dx}\left( \frac{u}{v} \right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \). We will use this formula to find the derivative of \( \frac{x}{\sin(x)} \).

Step 3: Compute the Derivative of the Numerator (\( u \))

The derivative of \( u = x \) with respect to \( x \) is \( \frac{du}{dx} = 1 \).

Step 4: Compute the Derivative of the Denominator (\( v \))

The derivative of \( v = \sin(x) \) with respect to \( x \) is \( \frac{dv}{dx} = \cos(x) \).

Step 5: Apply the Quotient Rule

Substitute the derivatives into the quotient rule formula: \[ \frac{d}{dx}\left( \frac{x}{\sin(x)} \right) = \frac{\sin(x) \cdot 1 - x \cdot \cos(x)}{\sin^2(x)}. \] Simplify to \( \frac{\sin(x) - x \cos(x)}{\sin^2(x)} \).

Key Concepts

Derivative

A derivative represents the rate of change of a function with respect to a variable. In calculus, taking a derivative is crucial for understanding how a function behaves as its input changes. When you look at a function, its derivative can tell you how steep the graph is at any point.

In our exercise, we are interested in the derivative of the function given by the expression \( x/\sin(x) \) with respect to \( x \). By calculating this derivative, we get insight into how the output of this function changes as \( x \) changes. It's like having a glimpse of the function's trend at any specific point.

Numerator and Denominator Identification

When you have a quotient of two functions, the first step in using the Quotient Rule is to clearly identify which part of the expression is the numerator and which part is the denominator.

For the expression \( x/\sin(x) \):

• The numerator is \( u = x \).
• The denominator is \( v = \sin(x) \).

This identification step is critical because we will separately take the derivatives of both the numerator and the denominator and then combine them using the Quotient Rule formula.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. In calculus, they often show up in various contexts, especially in derivations involving periodic functions or those expressed in terms of angles.

In this problem, the trigonometric function we encounter is \( \sin(x) \). Understanding its derivative is essential:

• The derivative of \( \sin(x) \) with respect to \( x \) is \( \cos(x) \).

This knowledge is part of fundamental calculus and helps compute the derivative of more complex expressions involving trigonometric terms.

Calculus Problem Solving

Solving calculus problems often involves a series of steps that systematically simplify complex expressions. By applying the rules of differentiation, such as the Quotient Rule, we make sense of how functions behave.

For \( \frac{x}{\sin(x)} \), the Quotient Rule formula is our tool:

• First, compute the derivatives of the numerator \( \frac{du}{dx} = 1 \) and the denominator \( \frac{dv}{dx} = \cos(x) \).
• Then, plug these into the Quotient Rule formula: \[ \frac{d}{dx}\left(\frac{x}{\sin(x)}\right) = \frac{\sin(x) \cdot 1 - x \cdot \cos(x)}{\sin^2(x)}. \]
• Finally, simplify the expression to get the result \( \frac{\sin(x) - x \cos(x)}{\sin^2(x)} \).

Calculus problem-solving is about breaking down complicated tasks into manageable parts, each step bringing us closer to understanding the function's behavior.