Problem 182 - Calculus Volume 1

Question

Find \(\frac{d y}{d x}\) for the given functions. $$ y=\frac{\tan x}{1-\sec x} $$

Step-by-Step Solution

Answer

Apply the quotient rule for \( y = \frac{\tan x}{1-\sec x} \) to find \( \frac{dy}{dx} = \frac{(1 - \sec x) \sec^2 x - \tan^2 x \sec x}{(1 - \sec x)^2} \).

Step 1: Identify the Function Structure

The given function is a rational function of two trigonometric expressions, where the numerator is \( \tan x \) and the denominator is \( 1 - \sec x \). It's structured as a quotient where we need to apply the quotient rule.

Step 2: Apply the Quotient Rule

To find \( \frac{dy}{dx} \) of the function \( y = \frac{u}{v} \), where \( u = \tan x \) and \( v = 1 - \sec x \), apply the quotient rule: \[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \]

Step 3: Calculate Derivatives of u and v

Find the derivatives of the numerator and denominator:- \( \frac{du}{dx} = \frac{d}{dx}(\tan x) = \sec^2 x \).- \( \frac{dv}{dx} = \frac{d}{dx}(1 - \sec x) = 0 + \sec x \tan x = \sec x \tan x \).

Step 4: Substitute Derivatives into Quotient Rule

Substitute the derivatives \( \frac{du}{dx} \) and \( \frac{dv}{dx} \) into the quotient rule formula:\[ \frac{dy}{dx} = \frac{(1 - \sec x) \cdot \sec^2 x - \tan x \cdot \sec x \tan x}{(1 - \sec x)^2} \]

Step 5: Simplify the Expression

Simplify the expression:- The numerator becomes \[ (1 - \sec x) \sec^2 x - \tan^2 x \sec x \]- Simplify further using trigonometric identities, if possible, to result in a more concise expression or to highlight any potential simplification.

Step 6: Final Expression for the Derivative

After simplification, the expression for \( \frac{dy}{dx} \) retains the structure, as trigonometric identities can vary based on context. However, ensure all calculations have been checked for accuracy and clarity.

Key Concepts

Quotient Rule

The quotient rule is a technique used in calculus to differentiate functions that are formed as a quotient of two other functions. When you have a function of the form \( y = \frac{u}{v} \), where both \( u \) and \( v \) are themselves functions of \( x \), the derivative \( \frac{dy}{dx} \) is calculated using the quotient rule formula:
\[ \frac{dy}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \]
This formula is derived from the product rule, and it helps you take the derivative of a fraction by considering the derivatives of the numerator and the denominator separately.

To apply the quotient rule correctly:

• Identify the numerator \( u \) and the denominator \( v \).
• Calculate their derivatives \( \frac{du}{dx} \) and \( \frac{dv}{dx} \).
• Substitute these into the quotient rule formula.
• Simplify the resulting expression if possible.

It’s essential to follow the steps systematically to avoid mistakes and find the correct derivative.

Trigonometric Derivatives

When differentiating functions involving trigonometric terms like \( \tan x \) and \( \sec x \), it's crucial to know their basic derivatives. Here’s a quick reminder:

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

These derivatives are foundational in calculus when dealing with trigonometric functions, particularly in expressions requiring the product, quotient, or chain rule.

For the given exercise, understanding these derivatives is key when applying the quotient rule. For \( y = \frac{\tan x}{1 - \sec x} \):

• The numerator \( u = \tan x \), and thus \( \frac{du}{dx} = \sec^2 x \).
• The denominator \( v = 1 - \sec x \), with \( \frac{dv}{dx} = \sec x \tan x \) (since the constant derivative is zero).

The accurate computation of these derivatives is crucial for finding the correct expression for \( \frac{dy}{dx} \).

Simplification of Derivatives

After applying the quotient rule and calculating the derivatives, the next crucial step is simplifying the resulting expression. This ensures that the derivative is presented in its simplest and most meaningful form. Simplification often involves rearranging terms and applying trigonometric identities.

In the problem, after substituting the derivatives into the quotient rule formula, you obtain a complex fraction:
\[ \frac{dy}{dx} = \frac{(1 - \sec x) \cdot \sec^2 x - \tan x \cdot \sec x \tan x}{(1 - \sec x)^2} \]

To simplify this:

• First, distribute terms within the numerator.
• Check if trigonometric identities like \( 1 + \tan^2 x = \sec^2 x \) can simplify the terms further.
• Combine like terms where possible to achieve the simplest form.

Although sometimes simplification doesn’t lead to a dramatically shorter expression, it helps clarify the relationship between the terms in the derivative. Be thorough in checking if any algebraic simplifications or factorizations are possible.