Problem 39 - University Calculus: Early Transcendentals

Question

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\tan ^{-1} \sqrt{x^{2}-1}+\csc ^{-1} x, \quad x>1$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

The function is given as \( y = \tan^{-1}(\sqrt{x^2-1}) + \csc^{-1}(x) \), and we need to find the derivative \( \frac{dy}{dx} \). This involves differentiating an inverse trigonometric function and an inverse cosecant function.

Step 2: Differentiate the Inverse Tangent Function

The derivative of \( \tan^{-1}(u) \) with respect to \(x\) is \( \frac{1}{1+u^2} \cdot \frac{du}{dx} \). For \( u = \sqrt{x^2-1} \), \( \frac{du}{dx} = \frac{x}{\sqrt{x^2-1}} \). Therefore, the derivative of \( \tan^{-1}(\sqrt{x^2-1}) \) is \( \frac{1}{1+(x^2-1)} \cdot \frac{x}{\sqrt{x^2-1}} = \frac{x}{x^2\sqrt{x^2-1}} \).

Step 3: Differentiate the Inverse Cosecant Function

The derivative of \( \csc^{-1}(x) \) is \( -\frac{1}{|x|\sqrt{x^2-1}} \). Since \( x > 1 \), we don't need to worry about the absolute value, so the derivative becomes \( -\frac{1}{x\sqrt{x^2-1}} \).

Step 4: Combine the Results

Add the derivatives obtained for each component of the function. \( \frac{dy}{dx} = \frac{x}{x^2\sqrt{x^2-1}} - \frac{1}{x\sqrt{x^2-1}} \).

Step 5: Simplify the Expression

Combine the fractions into a single fraction. The common denominator is \( x^2 \sqrt{x^2-1} \), so the expression simplifies to \[ \frac{x^2 - 1}{x^2 \sqrt{x^2-1}}. \]

Step 6: Final Derivative

The simplified derivative is \( \frac{x^2-1}{x^2\sqrt{x^2-1}} \).

Key Concepts

Understanding Derivatives

In calculus, a derivative represents the rate at which a function changes at a certain point. It gives us the slope of the tangent line to the function's graph at that specific point. Knowing derivatives helps in understanding how a function behaves, indicating where it increases, decreases, or has any extrema (peaks or valleys).

The notation for a derivative is usually written as \( \frac{dy}{dx} \), representing the derivative of \( y \) with respect to \( x \). This derivative tells us how \( y \) changes with small changes in \( x \). In the exercise, we are finding the derivative of a function composed of inverse trigonometric components.

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverse functions of the basic trigonometric functions like sin, cos, and tan. They help us find the angle whose trigonometric value is a given number. In our exercise, we are dealing with the inverse tangent (\( \tan^{-1} \)) and inverse cosecant (\( \csc^{-1} \)) functions.

• For \( \tan^{-1}(u) \), it provides the angle whose tangent is \( u \).
• For \( \csc^{-1}(x) \), it finds the angle whose cosecant is \( x \).

Understanding these functions is crucial in calculus, especially when differentiating them, as they often appear in the integration and differentiation of complex expressions.

Differentiation in Calculus

Differentiation is the process of finding a derivative. It is a fundamental tool in calculus, used to compute the rate of change of any given function. While differentiating, we apply various rules like the chain rule, product rule, and quotient rule to simplify the process.

In our exercise, we used the chain rule for differentiating both inverse trigonometric functions. This rule is useful when dealing with composite functions, where one function is nested inside another. For instance, \( \tan^{-1}(\sqrt{x^2-1}) \) involves the square root function inside the inverse tangent function, which requires applying the chain rule to successfully differentiate.

A Step-by-Step Solution Approach

A clear step-by-step solution is vital for understanding complex calculus problems. Here is how it was structured in our exercise:

• Step 1: Start by understanding the problem and the functions involved.
• Step 2: Decide on the method to differentiate the inverse tangent function, applying the chain rule as necessary.
• Step 3: Differentiate the inverse cosecant function, again using appropriate differentiation rules.
• Step 4: Combine the derivatives. Ensure all parts are combined in a single coherent expression.
• Step 5: Simplify the final expression. This often involves algebraic manipulation to present the derivative in its simplest form.

These structured steps not only solve the problem but also enhance the understanding of applying differentiation techniques to functions involving inverse trigonometric expressions.