Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \underline{\phantom{xxx}} \text { find } D_{x} y . $$ $$ y=\csc x=1 / \sin x $$

Step-by-Step Solution

Answer

The derivative \( D_x y \) is \( -\csc x \cot x \).

Step 1: Understand the Problem

You need to find the derivative of the function \( y = \csc x \) with respect to \( x \). Knowing that \( \csc x = \frac{1}{\sin x} \), we will use the concept of derivative to solve the problem.

Step 2: Recall the Derivative Rule for Cosecant

The derivative of \( \csc x \) is \( -\csc x \cot x \). This is a standard derivative that you should memorize for trigonometric functions.

Step 3: Differentiate Using the Result

Apply the derivative rule discussed in Step 2: \[D_x(\csc x) = - \csc x \cot x\]Therefore, the derivative \( D_x y \) is \( - \csc x \cot x \).

Step 4: Apply the Result to the Function

Substitute back into the original function: Since you have \( y = \csc x \), then \( D_x y = - \csc x \cot x \).

Key Concepts

Cosecant Function

The cosecant function, often abbreviated as "csc," is one of the lesser-known trigonometric functions. It is the reciprocal of the sine function, which means \( \csc x = \frac{1}{\sin x}\).
This function is important in calculus and trigonometry because of its special properties and relationships with other trig functions.

• The cosecant function is undefined wherever the sine function is zero, such as at angles like \(0\), \(\pi\), and \(2\pi\).
• This function is periodic, with a period of \(2\pi\).
• Cosecant's graph consists of repeating waves similar to sine, but inverted, appearing wherever sine has a local minimum or maximum.

Understanding these properties is essential for successfully differentiating and integrating it in a variety of mathematical contexts.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable where the functions are defined. They are crucial tools in calculus and algebra when simplifying expressions involving trigonometric terms.

• A basic identity involves the reciprocal nature of sine and cosecant: \(1 = \sin x \times \csc x\).
• Pythagorean identities, such as \(\sin^2 x + \cos^2 x = 1\), help simplify calculations and prove other identities.
• The quotient identities, like \(\tan x = \frac{\sin x}{\cos x}\), interlink different trigonometric functions.

These identities are often vital in calculus, particularly for converting complex expressions into something more manageable. Knowing these allows rapid transformation of trigonometric functions to solve derivatives more readily.

Calculus Differentiation

Differentiation is a fundamental concept in calculus that involves finding the rate at which a function is changing at any given point. For the function \(y = \csc x\), deriving it directly can feel daunting due to its reciprocal nature.
Use the standard derivative rules of trigonometric functions to streamline this process:

• The derivative of \(\sin x\) is \(\cos x\).
• The reciprocal rule states that if \(y = \frac{1}{f(x)}\), then \(y' = -\frac{f'(x)}{[f(x)]^2}\).
• Combining with the known formula: the derivative of \(\csc x\) is \(-\csc x \cot x\).

Recognizing these rules, you can more easily differentiate other trigonometric functions, ensuring you follow standard calculus practices when handling trigonometric derivatives.