In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
39. y=arctan√(x²-1) + arccsc(x), x>1

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Identify the function to differentiate: \(y = \arctan\left(\sqrt{x^{2} - 1}\right) + \arccsc(x)\), where \(x > 1\).

Recall the derivative formulas: For \(y = \arctan(u)\), \(\frac{dy}{dx} = \frac{1}{1 + u^{2}} \cdot \frac{du}{dx}\); for \(y = \arccsc(x)\), \(\frac{dy}{dx} = -\frac{1}{|x| \sqrt{x^{2} - 1}}\).

Differentiate the first term: Let \(u = \sqrt{x^{2} - 1} = (x^{2} - 1)^{1/2}\). Compute \(\frac{du}{dx}\) using the chain rule: \(\frac{du}{dx} = \frac{1}{2}(x^{2} - 1)^{-1/2} \cdot 2x = \frac{x}{\sqrt{x^{2} - 1}}\).

Apply the derivative formula for \(\arctan(u)\): \(\frac{d}{dx} \arctan\left(\sqrt{x^{2} - 1}\right) = \frac{1}{1 + (\sqrt{x^{2} - 1})^{2}} \cdot \frac{x}{\sqrt{x^{2} - 1}}\).

Differentiate the second term \(\arccsc(x)\) using its derivative formula: \(\frac{d}{dx} \arccsc(x) = -\frac{1}{|x| \sqrt{x^{2} - 1}}\). Since \(x > 1\), \(|x| = x\).

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주요 개념

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Derivative of Inverse Trigonometric Functions

Inverse trigonometric functions like arctan and arccsc have specific derivative formulas. For example, the derivative of arctan(u) is u' / (1 + u²), and the derivative of arccsc(x) is -1 / (|x|√(x² - 1)). Knowing these formulas is essential to differentiate the given function correctly.

Chain Rule

The chain rule is used to differentiate composite functions. When a function is composed of another function, such as arctan(√(x² - 1)), you first differentiate the outer function and then multiply by the derivative of the inner function. This rule is crucial for handling nested expressions.

Simplification of Derivatives Involving Radicals and Algebraic Expressions

After applying differentiation rules, simplifying expressions involving radicals and algebraic terms like √(x² - 1) is important. This often involves rationalizing denominators or combining terms to present the derivative in a clear, simplified form.

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.39. y=arctan√(x²-1) + arccsc(x), x>1