Problem 33 - Calculus Single Variable

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Functions

For the quotient rule, we identify the numerator and the denominator of the expression \( \frac{\sin(x)}{x} \). Here, \( u = \sin(x) \) and \( v = x \).

Step 2: Recall the Quotient Rule Formula

The quotient rule states that the derivative of \( \frac{u}{v} \) is given by \[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \].

Step 3: Find Derivatives of u and v

Compute the derivatives of \( u \) and \( v \). The derivative of \( u = \sin(x) \) is \( \frac{du}{dx} = \cos(x) \). The derivative of \( v = x \) is \( \frac{dv}{dx} = 1 \).

Step 4: Apply the Quotient Rule

Substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the quotient rule formula. We get: \[ \frac{d}{dx} \left( \frac{\sin(x)}{x} \right) = \frac{x \cdot \cos(x) - \sin(x) \cdot 1}{x^2} = \frac{x \cos(x) - \sin(x)}{x^2} \].

Step 5: Simplify the Expression

There is no further simplification needed for the expression \( \frac{x \cos(x) - \sin(x)}{x^2} \), as it is already in its simplest form.

Key Concepts

Derivative

A derivative represents how a function changes as its input changes. In calculus, it's all about finding the rate at which something changes. When you hear "derivative," think of it as the "instantaneous rate of change" or the "slope" of the function at a given point.

For the expression \( \frac{\sin(x)}{x} \), finding its derivative involves understanding how it behaves as \( x \) changes. Using the quotient rule helps us find this derivative efficiently.

Derivatives are basic building blocks for understanding trends over time in functions and are particularly useful in physics, engineering, and various fields where change and rate are important. They help us predict future behavior of systems or uncover patterns in data.

Trigonometric Functions

Trigonometric functions like sine and cosine are crucial in calculus, as they describe periodic phenomena. The function \( \sin(x) \) tells us how the y-coordinate of a point on a unit circle changes as it moves around the circle. It has well-known properties, such as periodicity and symmetry.

To derive \( \sin(x) \) with respect to \( x \), the result is \( \cos(x) \). This relationship is foundational in calculus and trigonometry, as these functions often appear in real-world applications like sound waves or oscillating circuits.

Being comfortable with these functions and their derivatives is key for solving complex calculus problems. Moreover, the tight connection between sine and cosine is frequently used, as seen in this problem, to find derivatives using the quotient rule.

Calculus Steps

Mastering calculus involves understanding the steps needed to solve problems methodically.

For the function \( \frac{\sin(x)}{x} \), following a clear framework like the quotient rule allows us to systematically find its derivative. Here's a quick summary of our approach:

• Identify the functions that form the numerator \( (\sin(x)) \) and denominator \( (x) \).
• Recall the quotient rule formula, which guides the derivative process for fractions of functions.
• Compute the derivatives individually, \( \cos(x) \) for \( \sin(x) \) and \( 1 \) for \( x \).
• Substitute these derivatives into the quotient rule formula.
• Finally, simplify the expression if needed, although it may already be in its simplest form.

Each step involves critical thinking and ensures the outcome is correct and reliable. Keeping these steps in mind not only aids in solving the present problem but also provides a framework to tackle similar problems in a manageable way.