Problem 64 - Calculus for Biology and Medicine

Question

Use the quotient rule to show that $$ \frac{d}{d x} \csc x=-\csc x \cot x $$

Step-by-Step Solution

Answer

The derivative of \( \csc x \) is \(-\csc x \cot x\).

Step 1: Define the Functions

Recall that the cosecant function is defined as the reciprocal of the sine function. Therefore, we can express the cosecant function as:\[ \csc x = \frac{1}{\sin x} \]

Key Concepts

Calculus

Calculus is a branch of mathematics that studies the rate of change of quantities. It primarily deals with limits, functions, derivatives, and integrals. Calculus is a powerful tool in understanding how changes occur over time or across different dimensions. Understanding calculus helps in many fields, like physics, engineering, economics, and more.

In this exercise, we focus on finding the derivative of the cosecant function using the quotient rule. The derivative, a core concept of calculus, represents the rate at which a function is changing at any given point. By mastering the application of rules like the quotient rule, one can solve complex differentiation problems efficiently.

Cosecant Function

The cosecant function, denoted as \( \csc x \), is the reciprocal of the sine function. This means:

• \( \csc x = \frac{1}{\sin x} \).
• It is undefined when \( \sin x = 0 \), which occurs at integer multiples of \( \pi \).
• The function is periodic with a period of \( 2\pi \).

Understanding the properties of \( \csc x \) is crucial in calculus, as these properties inform us about where the function has asymptotes and help in determining its domain and range. Additionally, knowing that \( \csc x \) is the reciprocal of a basic trigonometric function allows us to apply calculus's differentiation rules, such as the quotient rule, effectively.

Derivative

The derivative of a function quantifies its rate of change or slope at any given point. When working with trigonometric functions, derivatives help in understanding how these functions behave as the input values change.

For the reciprocal trigonometric function \( \csc x = \frac{1}{\sin x} \), we use the quotient rule to find its derivative. The quotient rule is:

• If you have a function \( y = \frac{u}{v} \), then \( \frac{dy}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \).

In our exercise, set \( u = 1 \) and \( v = \sin x \). Thus, \( u' = 0 \) and \( v' = \cos x \). Applying the quotient rule, the derivative of \( \csc x \) becomes:

• \( \frac{d}{dx} \csc x = \frac{(\sin x)\cdot 0 - 1\cdot \cos x}{(\sin x)^2} \).
• \( = -\frac{\cos x}{\sin^2 x} \).
• By recognizing that \( \frac{\cos x}{\sin^2 x} = \csc x \cot x \), the derivative results in \( -\csc x \cot x \).