Problem 67 - Thomas Calculus

Question

In Exercises \(49-70\) , 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

\( \frac{dy}{dx} = \frac{x - 1}{x^2 \sqrt{x^2 - 1}} \).

Step 1: Differentiate the Inverse Trigonometric Functions

To find the derivative of the function \( y = \tan^{-1} \sqrt{x^2 - 1} + \csc^{-1} x \), we first differentiate each term separately. For the first term, use the chain rule. The derivative of \( \tan^{-1} u \) is \( \frac{1}{1 + u^2} \), where \( u = \sqrt{x^2 - 1} \). We also need the derivative of \( u = \sqrt{x^2 - 1} \), which is \( \frac{x}{\sqrt{x^2 - 1}} \). For the second term, the derivative of \( \csc^{-1} x \) is \( -\frac{1}{|x|\sqrt{x^2 - 1}} \).

Step 2: Apply the Chain Rule to the First Term

Using the chain rule, differentiate \( \tan^{-1} \sqrt{x^2 - 1} \) as follows:\( \frac{d}{dx}\left( \tan^{-1} \sqrt{x^2 - 1} \right) = \frac{1}{1 + (\sqrt{x^2 - 1})^2} \cdot \frac{d}{dx} \left( \sqrt{x^2 - 1} \right) = \frac{1}{1 + (x^2 - 1)} \cdot \frac{x}{\sqrt{x^2 - 1}} = \frac{x}{x^2 \sqrt{x^2 - 1}} \).

Step 3: Differentiate the Second Term

For \( \csc^{-1} x \), the derivative is:\( \frac{d}{dx} \left( \csc^{-1} x \right) = -\frac{1}{|x|\sqrt{x^2-1}} \). Given that \( x > 1 \), \( |x| = x \). So, it simplifies to \( -\frac{1}{x\sqrt{x^2-1}} \).

Step 4: Combine the Results

Add the derivatives from Step 2 and Step 3:\( \frac{dy}{dx} = \frac{x}{x^2 \sqrt{x^2 - 1}} - \frac{1}{x \sqrt{x^2 - 1}} \).

Step 5: Simplify the Expression

Combine the terms over a common denominator:\( \frac{dy}{dx} = \frac{x - 1}{x^2 \sqrt{x^2 - 1}} \). The common denominator is \( x^2 \sqrt{x^2 - 1} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions help us find angles when given certain ratios, essentially reversing the trigonometric functions. For example, if you know the sine of an angle, you can find the angle using inverse sine, or arcsin. In this exercise, we deal with \(\tan^{-1}\) (arctangent) and \(\csc^{-1}\) (arccosecant). These functions are used to indicate the angles where their respective trigonometric functions would return the values given within their specific domains.

To differentiate these inverse functions correctly, we rely on specific derivative formulas. The derivative of \(\tan^{-1}(u)\) is \(\frac{1}{1+u^2}\) while the derivative of \(\csc^{-1}(x)\) is \(-\frac{1}{|x|\sqrt{x^2 - 1}}\).

• \(\tan^{-1}(u)\) adjusts for angle values using tangent ratios.
• \(\csc^{-1}(x)\) deals with reciprocal sine values, evaluating proper angle values.

By understanding these derivatives, we can explore more complex functions involving inverse trigonometric identities.

Chain Rule

When a function is wrapped inside another function, we use the chain rule to differentiate it. The chain rule is a powerful technique used to find the derivative of composite functions. The basic idea is to take the derivative of the outer function and multiply it by the derivative of the inner function.

For example, with the expression \(\tan^{-1}( \sqrt{x^2 - 1} )\), \(\sqrt{x^2 - 1}\) is our inner function. To differentiate using the chain rule, we first find the derivative of \(\tan^{-1}(u)\) with respect to \(u\), giving us \(\frac{1}{1+u^2}\). Then, we multiply this result by the derivative of \(\sqrt{x^2 - 1}\).

• Find the derivative of \(u\): \(\frac{d}{dx}(\sqrt{x^2 - 1}) = \frac{x}{\sqrt{x^2 - 1}}\).
• Chain the derivatives for \(\tan^{-1}(u)\).

This rule is helpful because it allows us to systematically approach and solve complex derivative problems, especially when dealing with layers of nested functions.

Derivative Calculations

Calculating derivatives lets us understand how functions change, how steep their graphs are, or predict trends in data. This exercise aims to differentiate a sum of inverse trigonometric functions. It boils down to applying derivative rules and simplifying the results.

The derivative of the given function \(y = \tan^{-1} \sqrt{x^2 - 1} + \csc^{-1} x \) involves calculating each part separately. After finding individual derivatives, they need combining into a neat solution. This involves:

• Using common denominators to combine terms, like finding a common base for \(x\) and handling square roots.
  
• Simplifying the equation to \(\frac{x - 1}{x^2 \sqrt{x^2 - 1}}\).

Doing so, we ensure a cleaner, easier-to-understand derivative that reflects both the change rates and angles involved. Derivative calculations help highlight relationships between components of mathematical expressions by showing us how they vary relative to one another.