Problem 28 - University Calculus: Early Transcendentals

Question

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\csc ^{-1} \frac{x}{2}$$

Step-by-Step Solution

Answer

The derivative is \(-\frac{2}{|x| \sqrt{x^2 - 4}}\).

Step 1: Recognize the Function Type

The given function is an inverse cosecant function, specifically it is \( y = \csc^{-1}(\frac{x}{2}) \). Inverse trigonometric functions require specific differentiation rules.

Step 2: Apply the Derivative Formula

The derivative of \( y = \csc^{-1}(u) \) with respect to \( x \) is \( \frac{dy}{dx} = -\frac{1}{|u| \sqrt{u^2 - 1}} \cdot \frac{du}{dx} \), where \( u \) is a function of \( x \). Here, \( u = \frac{x}{2} \).

Step 3: Differentiate the Inner Function

Differentiating \( u = \frac{x}{2} \) with respect to \( x \), we get \( \frac{du}{dx} = \frac{1}{2} \).

Step 4: Substitute Back into the Derivative Formula

Substitute \( u = \frac{x}{2} \) and \( \frac{du}{dx} = \frac{1}{2} \) back into the derivative formula. We get: \( \frac{dy}{dx} = -\frac{1}{|\frac{x}{2}| \sqrt{(\frac{x}{2})^2 - 1}} \cdot \frac{1}{2} \).

Step 5: Simplify the Expression

First, simplify the magnitude of \( \frac{x}{2} \) as \( \frac{|x|}{2} \). Then simplify the expression under the square root: \((\frac{x}{2})^2 = \frac{x^2}{4}\), thus \(\frac{x^2}{4} - 1 = \frac{x^2 - 4}{4}\). Therefore, \( \sqrt{\frac{x^2}{4} - 1} = \frac{\sqrt{x^2 - 4}}{2} \).

Step 6: Finalize the Derivative Expression

Substituting these into the derivative, we get: \( \frac{dy}{dx} = -\frac{1}{\frac{|x|}{2} \cdot \frac{\sqrt{x^2 - 4}}{2}} \cdot \frac{1}{2} \) which simplifies to \( \frac{dy}{dx} = -\frac{2}{|x| \sqrt{x^2 - 4}} \).

Key Concepts

Inverse Cosecant Function

The inverse cosecant function, denoted as \( \csc^{-1}(x) \), is the inverse of the cosecant function. It is one of the six inverse trigonometric functions. This function provides the angle whose cosecant is \( x \). The domain of the arcsine function and its combinations, like inverse cosecant, is limited to values where the cosecant function is defined and can yield a real number.
- The range of the inverse cosecant function lies between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), excluding zero.- As with other inverse trigonometric functions, typically only certain intervals are used to ensure the function is one-to-one and thus invertible.
When differentiating expressions involving inverse cosecant, it begins with recognizing the characteristic inverse function, which then applies specific differentiation rules.

Differentiation Rules

Differentiation rules provide us with a structured way of finding the derivative of a function. These rules include basic rules like the power rule, product rule, quotient rule, and chain rule. For inverse functions, like inverse trigonometric functions, special differentiation rules exist.
- The chain rule is essential when dealing with inverse functions. This rule helps us differentiate compositions of functions, such as \( \csc^{-1}\left( \frac{x}{2} \right) \) where an inner function \( u = \frac{x}{2} \) is involved.
- Using these rules, we can express the derivative of any function concisely and accurately. Recognizing when to use each rule is crucial, as it simplifies complex differentiation problems and makes the solution straightforward.

Trigonometric Differentiation

Trigonometric differentiation involves finding the derivative of trigonometric functions and their inverses. It's important in calculus due to the frequent appearance of trigonometric functions in various problems.
- Regular trigonometric functions (sine, cosine, tangent, etc.) have straightforward differentiation rules. However, inverse functions like \( \csc^{-1} \), \( \sin^{-1} \), etc., require specific formulas to determine derivatives.
- For instance, the derivative of \( y = \csc^{-1}(x) \) with respect to \( x \) is given by the formula: \( \frac{dy}{dx} = -\frac{1}{|x| \sqrt{x^2 - 1}} \).
Using these formulas allows us to manage problems involving inverse trigonometric functions more effectively.

Inverse Function Differentiation

Differentiating inverse functions, particularly inverse trigonometric functions, requires a clear understanding of their derivatives. The general formula takes into account the absolute value of the input to handle restrictions in their domain.
- The derivative of an inverse trigonometric function \( y = \csc^{-1}(u) \) is calculated using the formula \( \frac{dy}{dx} = -\frac{1}{|u| \sqrt{u^2 - 1}} \cdot \frac{du}{dx} \), where \( u \) depends on \( x \).
- This emphasizes the importance of the chain rule, as it allows the differentiation of complex functions by breaking them into simpler components.
When given an expression involving inverse functions, carefully identify the inner function and apply the specific differentiation formula to find derivatives. This method is reliable and simplifies otherwise complex operations.