Problem 182 - Calculus Volume 1

Question

For the following exercises, find \(\frac{d y}{d x}\) for the given functions. $$y=\frac{\tan x}{1-\sec x}$$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = \frac{\sec^2 x - \sec^3 x + \sec x \tan^2 x}{(1 - \sec x)^2} \)

Step 1: Identify the Function Structure

The given function is \( y = \frac{\tan x}{1 - \sec x} \). It is a quotient. To differentiate it, we will use the Quotient Rule, which states: \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \) where \( u = \tan x \) and \( v = 1 - \sec x \).

Step 2: Differentiate the Numerator

Find the derivative of the numerator \( u = \tan x \). The derivative is \( \frac{du}{dx} = \sec^2 x \).

Step 3: Differentiate the Denominator

Find the derivative of the denominator \( v = 1 - \sec x \). The derivative is \( \frac{dv}{dx} = 0 - (\sec x \tan x) = -\sec x \tan x \).

Step 4: Apply the Quotient Rule

Substitute \( \frac{du}{dx} \), \( \frac{dv}{dx} \), \( u \), and \( v \) into the Quotient Rule formula: \[ \frac{d}{dx} \left( \frac{\tan x}{1 - \sec x} \right) = \frac{(1 - \sec x) \cdot \sec^2 x - \tan x \cdot (-\sec x \tan x)}{(1 - \sec x)^2} \].

Step 5: Simplify the Expression

Simplify the expression in the numerator:- The first part is \((1 - \sec x) \cdot \sec^2 x = \sec^2 x - \sec^3 x\).- The second part is \(-\tan x \cdot (-\sec x \tan x) = \sec x \tan^2 x\).Combine these to get \( \sec^2 x - \sec^3 x + \sec x \tan^2 x \).

Step 6: Final Formulation

Combine everything to write the derivative:\[ \frac{dy}{dx} = \frac{\sec^2 x - \sec^3 x + \sec x \tan^2 x}{(1 - \sec x)^2} \].

Key Concepts

quotient rule

The quotient rule is a fundamental technique in calculus to find the derivative of a division of two functions. When you have a function that is a fraction, where one function is divided by another (like our case, \(y = \frac{\tan x}{1-\sec x}\)), the quotient rule is used to differentiate it.
The formula for the quotient rule is \[\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}\]. Here, \(u\) is the nominator and \(v\) is the denominator of the fraction.

To effectively use the quotient rule, follow these steps:

• Identify the top function \(u\) and the bottom function \(v\).
• Find the derivative of \(u\) with respect to \(x\), denoted as \(\frac{du}{dx}\).
• Find the derivative of \(v\) with respect to \(x\), denoted as \(\frac{dv}{dx}\).
• Substitute \(u\), \(v\), \(\frac{du}{dx}\), and \(\frac{dv}{dx}\) into the quotient rule formula.
• Simplify the resulting expression to find the derivative.

This approach helps in tackling more complex functions that are presented as fractions, especially those involving trigonometric expressions.

trigonometric functions

Trigonometric functions are crucial in calculus, providing the tools to model periodic behavior. Common functions include \(\sin x\), \(\cos x\), \(\tan x\), \(\csc x\), \(\sec x\), and \(\cot x\). In our exercise, the function \(\tan x\) and \(\sec x\) play significant roles.
The derivative of each trigonometric function follows a specific rule:

• The derivative of \(\tan x\) is \(\sec^2 x\).
• The derivative of \(\sec x\) is \(\sec x \tan x\).

These derivatives are useful as they help simplify the process of differentiation when dealing with larger expressions involving these functions.

The exercise uses these specific derivatives when applying the quotient rule. It's also important to recognize how these derivatives interact when combined.
Understanding these basic derivatives allows you to manage more complex calculus problems involving periodic motions or wave forms.

differentiation techniques

Differentiation techniques in calculus involve methods to find the rate of change of a function with respect to a variable. These techniques include basic rules, such as the power rule, product rule, chain rule, and importantly for this problem, the quotient rule. Each technique corresponds to a different type of function's structure.
In the given exercise, differentiation was done using the quotient rule since the function involves division. Different contexts would necessitate different techniques, but the quotient rule is optimal when dealing with a numerator and a denominator as separate functions.

The differentiation process involves a few steps:

• Identifying the type of function (e.g., product of two functions, quotient, composite function).
• Applying the appropriate rule (such as power rule, chain rule, etc.).
• Calculating the derivative of each component involved in the expression.
• Simplifying the result to achieve a readable and manageable derivative expression.

By mastering these techniques, you are equipped to handle various mathematical challenges involving derivatives in calculus. Each technique, including those mentioned, fits a specific situation, allowing you to analyze complex functions and their rates of change.