Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problema 7.6.39

Capitolo 7, Problema 7.6.39

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

Guida verificata passo dopo passo

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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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.

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Derivatives of Other Inverse Trigonometric Functions

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.

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Intro to the Chain Rule

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.

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Integrals Involving Natural Logs: Substitution Example 8

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