Problem 179 - Calculus Volume 1

Question

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

Step-by-Step Solution

Answer

The derivative is \( \frac{\sec x (x \tan x - 1)}{x^2} \).

Step 1: Identify the Differentiation Rule

The given function is a quotient of two functions, \( y = \frac{\sec x}{x} \). To differentiate it, we will use the Quotient Rule, which is defined for two functions \( u(x) \) and \( v(x) \) as \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2} \). Here, \( u = \sec x \) and \( v = x \).

Step 2: Differentiate the Numerator

Find the derivative of the numerator \( u = \sec x \). The derivative \( u' = \sec x \tan x \).

Step 3: Differentiate the Denominator

Find the derivative of the denominator \( v = x \). The derivative \( v' = 1 \).

Step 4: Apply the Quotient Rule

Now apply the Quotient Rule: \[ \frac{d}{dx} \left( \frac{\sec x}{x} \right) = \frac{x \cdot \sec x \tan x - \sec x \cdot 1}{x^2} \].

Step 5: Simplify the Expression

Simplify the expression obtained from the Quotient Rule:\[ \frac{x \sec x \tan x - \sec x}{x^2} = \frac{\sec x (x \tan x - 1)}{x^2} \].

Key Concepts

Quotient Rule

When we have a function that is a division of two functions, we use a technique called the Quotient Rule to differentiate it. The Quotient Rule is very helpful because it gives us a structured way to handle these types of problems, which might otherwise be quite tricky. Suppose you have a function given by the quotient of two functions, let's call them \( u(x) \) and \( v(x) \), so the function is \( \frac{u}{v} \). The Quotient Rule formula tells us how to differentiate this:

• First, identify the functions \( u \) and \( v \) from the quotient. Here, it's \( u = \sec x \) and \( v = x \).
• Then, calculate their derivatives: \( u' = \sec x \tan x \) and \( v' = 1 \).
• Finally, apply the formula: \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2} \).

For our example, this results in: \( \frac{d}{dx} \left( \frac{\sec x}{x} \right) = \frac{x \cdot \sec x \tan x - \sec x}{x^2} \). This approach makes it easier to manage the complex operation of differentiation in fractions.

Trigonometric Differentiation

Trigonometric functions, like \( \sec x \), \( \tan x \), and others, have specific derivatives that you can memorize to make differentiation quicker and more efficient. This particular function, \( y = \frac{\sec x}{x} \), employs the differentiation of \( \sec x \). Knowing the derivatives of these basic trigonometric functions is crucial because it allows you to focus on applying rules like the Quotient Rule without getting stuck on basic derivatives.Key derivatives to remember include:

• \( \frac{d}{dx}(\sin x) = \cos x \)
• \( \frac{d}{dx}(\cos x) = -\sin x \)
• \( \frac{d}{dx}(\tan x) = \sec^2 x \)
• \( \frac{d}{dx}(\sec x) = \sec x \tan x \)

By memorizing these fundamental derivatives, you'll be well-prepared to tackle more complex problems involving trigonometric differentiation.

Derivatives of Trigonometric Functions

Trigonometric functions are widely used in calculus, and each has a specific derivative that you need to learn. These derivatives not only apply to the primary trigonometric functions like sine and cosine but extend to their reciprocal functions like secant, cosecant, tangent, and cotangent.Here’s a quick list of some common trigonometric derivatives for reference:

• The derivative of \( \sin x \) is \( \cos x \).
• The derivative of \( \cos x \) is \(-\sin x \).
• The derivative of \( \tan x \) is \( \sec^2 x \).
• The derivative of \( \sec x \) is \( \sec x \tan x \), which we used in this particular example.
• The derivative of \( \csc x \) is \(-\csc x \cot x \).
• The derivative of \( \cot x \) is \(-\csc^2 x \).

It’s important to practice these derivatives regularly because they are foundational to many topics in calculus. These derivatives are tools that will enable you to solve diverse problems across different calculus applications.