Problem 33 - University Calculus: Early Transcendentals

Question

Find \(y^{\prime \prime}\) if a. \(y=\csc x\) b. \(y=\sec x\)

Step-by-Step Solution

Answer

For a, \(y'' = \csc^3 x + \csc x \cot^2 x\). For b, \(y'' = \sec x (\sec^2 x + \tan^2 x)\).

Step 1: Differentiate the function y = csc(x) for the first derivative

To find the first derivative of the function \(y = \csc x\), apply the differentiation formula for the cosecant function: \( \frac{d}{dx}(\csc x) = -\csc x \cot x \). Thus, the first derivative is: \( y' = -\csc x \cot x \).

Step 2: Differentiate y' to find y'' for y = csc(x)

Differentiate the first derivative \(y' = -\csc x \cot x\) using the product rule, \( \frac{d}{dx}(uv) = u'v + uv' \). Here, \(u = \csc x\) and \(v = \cot x\). Find \(u' = -\csc x \cot x\) and \(v' = -\csc^2 x\). Therefore: \[ y'' = -(\csc x)(-\csc^2 x) - (\csc x \cot x)(\cot x)\]Simplify the terms to find the second derivative: \( y'' = \csc^3 x + \csc x \cot^2 x\).

Step 3: Differentiate the function y = sec(x) for the first derivative

To find the first derivative of the function \(y = \sec x\), use the differentiation formula for the secant function: \( \frac{d}{dx}(\sec x) = \sec x \tan x \). So, the first derivative is: \( y' = \sec x \tan x \).

Step 4: Differentiate y' to find y'' for y = sec(x)

To find the second derivative \(y''\), differentiate the first derivative \(y' = \sec x \tan x\) using the product rule. Let \(u = \sec x\) and \(v = \tan x\). Then \(u' = \sec x \tan x\) and \(v' = \sec^2 x\). Now differentiate: \[y'' = (\sec x)(\sec^2 x) + (\sec x \tan x)(\tan x)\]Simplify the resulting expression to find the second derivative: \(y'' = \sec x \sec^2 x + \sec x \tan^2 x\), which simplifies to \(y'' = \sec x (\sec^2 x + \tan^2 x)\).

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \ ext{csc} \, x \), is the reciprocal of the sine function. It plays a crucial role in trigonometry and calculus differentiation. When we discuss the cosecant function, we are focusing on the value \( \csc x = rac{1}{\sin x} \). This function has some distinctive properties:

• It is undefined wherever the sine function is zero, leading to vertical asymptotes.
• The range of the cosecant function consists of all real numbers except between -1 and 1.

Understanding the calculus aspect involves differentiating this function. The derivative of \(y = \csc x\) uses the formula: \( \frac{d}{dx}(\csc x) = -\csc x \cot x\). Differentiating once more provides the second derivative, exemplified by applying the product rule to find: \( y'' = \csc^3 x + \csc x \cot^2 x\). This detailed process ensures thorough comprehension and accurate calculation in further applications.

Secant Function

The secant function, represented by \( \ ext{sec} \, x \), is essential in both geometry and calculus. It is the reciprocal of the cosine function, mathematically expressed as \( \sec x = \frac{1}{\cos x} \). Key characteristics of the secant function include:

• Undefined where cosine is zero, leading to vertical asymptotes similar to the cosecant function.
• Its range is also all real numbers except for values between -1 and 1.

For differentiation, the first derivative of \(y = \sec x\) involves the formula: \( \frac{d}{dx}(\sec x) = \sec x \tan x\). Calculating the second derivative is done through further differentiation, utilizing the product rule to derive: \( y'' = \sec x \(\sec^2 x + \tan^2 x\)\). Mastery of these derivatives is essential for solving complex trigonometric calculus problems.

Second Derivative

The second derivative of a function offers insight into the function's curvature, revealing important features such as concavity and points of inflection. Specifically, for trigonometric functions, like those involving secant and cosecant, the second derivative provides a deeper understanding of behavior beyond the primary rate of change.
Understanding how to derive a second derivative involves these steps:

• The first step is taking the initial derivative of the function, applying relevant differentiation rules depending on the trigonometric function.
• The second derivative is calculated by differentiating the first derivative, using product rule or chain rule as necessary.

Through this, we deduce critical information about the function's geometric properties. For instance, the second derivative of the secant function \(\ y'' = \sec x (\sec^2 x + \tan^2 x) \) and the cosecant function \(\ y'' = \csc^3 x + \csc x \cot^2 x \) reveal how these functions bend and shift across the graph, leading to comprehensive insights essential for advanced calculus analysis.